# shortfx - Function Catalogue for LLMs

> shortfx is a Python library with 3000+ reusable functions organized like Excel formulas.
> PyPI package: `shortfx` (`pip install shortfx`). Import as `import shortfx`.
> This file helps LLMs discover function signatures and descriptions without calling a server.

## How to Use

Each function is listed as:
`
module.function_name(param1, param2, ...) -> description
`

**Total functions: 2932**

## fxDate.date_convertions

- \ad_format_to_datetime(p_ad_date_str: string)\ - Convierte una cadena de fecha/hora en formato 'YYYYMMDDHHMMSS.0Z' (usado en Active Directory)
- \convert_timezone(dt: any, from_tz: string, to_tz: string)\ - Converts a datetime between two IANA timezones.
- \date_to_excel_serial(d: any)\ - Converts a Python date to an Excel serial date number.
- \date_to_excel_timestamp(dt_val: any)\ - Converts a datetime to an Excel serial number including fractional day.
- \date_to_human_short(dt_value: any, language: string = en)\ - Formats a date as a short human-readable string.
- \date_to_iso_format(date_input: any, input_format: string = None)\ - Converts a datetime object to an ISO 8601 formatted string.
- \date_to_iso_week_date(d: any)\ - Convert a date to an ISO 8601 week date string.
- \date_to_julian(date_input: any, input_format: string = None)\ - Convierte una fecha estándar a su representación de fecha Juliana (día del año).
- \date_to_string(date_input: any, format_code: string = %Y-%m-%d)\ - Converts a datetime object into a formatted string.
- \datetime_to_ad_format(p_dt_input: any)\ - Convierte un objeto datetime o date a la cadena de formato
- \datetime_to_cron(dt_input: any)\ - Convert a datetime to a cron schedule expression.
- \datetime_to_date(datetime_obj: any)\ - Converts a datetime object to a date object, discarding time components.
- \datetime_to_filetime(p_dt: any)\ - Convierte un objeto datetime o date de Python a un valor FILETIME de Windows.
- \datetime_to_integer(date_input: any, input_format: string = None)\ - Converts a date to an integer in YYYYMMDD format, ignoring time.
- \datetime_to_iso8601(dt: any)\ - Converts a date or datetime to an ISO 8601 string.
- \datetime_to_milliseconds_timestamp(date_input: any)\ - Convierte un objeto datetime a un sello de tiempo Unix en milisegundos.
- \datetime_to_rfc2822(dt_input: any)\ - Format a datetime as an RFC 2822 string (email header format).
- \datetime_to_timestamp(date_input: any)\ - Converts a datetime object to a Unix timestamp.
- \decimal_hours_to_time(hours: number)\ - Converts decimal hours to a ``time`` object.
- \excel_serial_to_date(serial: integer)\ - Converts an Excel serial date number to a Python date.
- \filetime_to_datetime(filetime: integer)\ - Converts a Windows FILETIME value to a datetime object and its string representation.
- \from_iso_to_local_datetime(iso_string: string)\ - Converts an ISO 8601 formatted string to a datetime object in the local timezone.
- \hms_to_seconds(hms: string)\ - Parses an ``H:MM:SS`` or ``M:SS`` string to total seconds.
- \integer_to_datetime(yyyymmdd: integer)\ - Converts a YYYYMMDD integer back to a datetime object.
- \iso8601_to_datetime(text: string)\ - Parses an ISO 8601 string to a date or datetime object.
- \iso_week_date_to_date(text: string)\ - Parse an ISO 8601 week date string to a date object.
- \julian_to_date(julian_date: integer, year: integer)\ - Convierte una fecha Juliana (día del año) a un objeto datetime estándar.
- \list_available_timezones()\ - Retrieves and returns a sorted list of all available IANA time zone names.
- \milliseconds_timestamp_to_datetime(timestamp_ms: integer)\ - Convierte un sello de tiempo Unix en milisegundos a un objeto datetime.
- \modified_julian_date(d: any)\ - Convert a date to Modified Julian Date (MJD).
- \parse_date_flexible(text: string, dayfirst: boolean = True)\ - Parses a date string by trying multiple common formats.
- \rata_die(d: any)\ - Convert a date to Rata Die (absolute day number).
- \rfc2822_to_datetime(text: string)\ - Parse an RFC 2822 date string to a datetime object.
- \seconds_to_hms(seconds: integer)\ - Converts total seconds to an ``H:MM:SS`` string.
- \time_to_day_fraction(time_text: string)\ - Convert a time string to a day fraction (0.0 – 1.0).
- \time_to_decimal_hours(t: any)\ - Converts a ``time`` object to decimal hours.
- \time_zone_offset(tz_name: string, dt_val: any = None)\ - Returns the UTC offset string for a timezone at a given datetime.
- \timestamp_to_datetime(timestamp_input: integer)\ - Converts a Unix timestamp back to a datetime object.
- \unix_epoch_day(d: any)\ - Returns the ordinal day number since the Unix epoch (1970-01-01 = day 0).
- \unix_epoch_days(d: any)\ - Days elapsed since the Unix epoch (1970-01-01).
- \unix_timestamp_to_datetime(timestamp: integer, tz_info: string = None)\ - Converts a UNIX timestamp (seconds since the Epoch) to a datetime object.
- \utc_to_datetime(date_input: any, input_format: string = None, input_tz: string = None)\ - Converts any datetime object (aware or naive) or date string to its equivalent UTC representation.
- \utc_to_midnight_iso(p_datetime: any)\ - Convierte un datetime a su equivalente a medianoche UTC.
- \utc_to_timezone(input_datetime: any, target_tz_name: string = Europe/Madrid)\ - Localizes a naive datetime object to the target timezone or validates an aware datetime,

## fxDate.date_evaluations

- \age_at_date(birth_date: any, reference: any = None)\ - Calculate age in complete years at a reference date.
- \astronomical_season(d: any, hemisphere: string = north)\ - Determine the astronomical season based on equinox/solstice dates.
- \bimester_of_date(d: any)\ - Return the bimester number for a date.
- \century_of_date(d: any)\ - Return the century number of a date.
- \chinese_zodiac(year: integer)\ - Returns the Chinese zodiac animal for a given year.
- \date_grade(d: any)\ - Classify a date into a period label: 'ancient', 'medieval', 'modern', or 'contemporary'.
- \date_of_easter(year: integer)\ - Calculate the date of Easter Sunday using the Anonymous Gregorian algorithm.
- \date_of_nth_weekday(year: integer, month: integer, weekday: integer, n: integer)\ - Return the date of the *n*-th occurrence of a weekday in a month.
- \date_to_julian_day(d: any)\ - Return the Julian Day Number for a date.
- \day_name_of_date(d: any)\ - Return the English weekday name for a date.
- \days_in_year(year: integer)\ - Return the number of days in a given year (365 or 366).
- \days_until_end_of_year(d: any)\ - Return the number of days remaining until the end of the year.
- \days_until_next_birthday(birth_date: any, reference: any = None)\ - Calculate the number of days until the next birthday.
- \days_until_weekday(d: any, target_weekday: integer)\ - Return the number of days from *d* until the next occurrence of *target_weekday*.
- \elapsed_years(start: any, end: any)\ - Return the number of full years between two dates.
- \fiscal_quarter(d: any, fiscal_start_month: integer = 1)\ - Return the fiscal quarter (1-4) for a given date.
- \fortnight_of_year(d: any)\ - Return the fortnight number of the year (1-based).
- \generation_name(year: integer)\ - Return the generational cohort name for a birth year.
- \is_anniversary(reference: any, target: any)\ - Checks if target falls on the same month and day as reference.
- \is_blue_moon(year: integer, month: integer)\ - Check whether a given month contains a blue moon.
- \is_business_hours(dt: any, start_hour: integer = 9, end_hour: integer = 17)\ - Checks if a datetime falls within business hours on a weekday.
- \is_century_year(year: integer)\ - Check if a year is a century year (divisible by 100).
- \is_date_in_range(d: any, start: any, end: any)\ - Check if a date falls within [start, end] inclusive.
- \is_dateclass(p_datetime: any)\ - Checks if the provided object is a datetime instance.
- \is_dst(dt: any, tz_name: string = UTC)\ - Checks if a datetime falls within Daylight Saving Time.
- \is_dst_transition_day(d: any, tz_name: string)\ - Returns True if the given date has a DST (Daylight Saving Time) transition.
- \is_end_of_month(d: any)\ - Check if a date is the last day of its month.
- \is_end_of_quarter(d: any)\ - Check if a date falls on the last day of a calendar quarter.
- \is_equinox_or_solstice(d: any)\ - Return the astronomical event name if *d* falls on one, else ``None``.
- \is_first_day_of_month(d: any)\ - Check if a date is the first day of its month.
- \is_first_of_month(d: any)\ - Check if a date falls on the first day of the month.
- \is_friday_13th(d: any)\ - Check whether a date falls on Friday the 13th.
- \is_future(date_input: any)\ - Checks if a date is strictly in the future.
- \is_golden_hour(d: any, latitude: number, longitude: number)\ - Approximate whether a datetime falls within the golden hour.
- \is_holiday(d: any, country: string = ES)\ - Check if a date is a public holiday in the given country.
- \is_iso_long_year(year: integer)\ - Check if an ISO year has 53 weeks.
- \is_last_day_of_month(d: any)\ - Checks if a date is the last day of its month.
- \is_last_day_of_year(d: any)\ - Check if a date is December 31st.
- \is_millennium_year(year: integer)\ - Check if a year is a millennium year (divisible by 1000).
- \is_nth_weekday(d: any, n: integer, weekday: integer)\ - Check if *d* is the n-th occurrence of a weekday in its month.
- \is_palindrome_date(d: any)\ - Check if a date is a palindrome in YYYYMMDD format.
- \is_past(date_input: any)\ - Checks if a date is strictly in the past.
- \is_same_day(date1: any, date2: any)\ - Checks if two dates fall on the same calendar day.
- \is_same_month(date1: any, date2: any)\ - Checks if two dates fall in the same month and year.
- \is_same_quarter(date1: any, date2: any)\ - Checks if two dates fall in the same quarter and year.
- \is_same_week(date1: any, date2: any)\ - Checks if two dates fall in the same ISO week and year.
- \is_same_year(date1: any, date2: any)\ - Checks if two dates fall in the same year.
- \is_start_of_quarter(d: any)\ - Check if a date falls on the first day of a calendar quarter.
- \is_today(date_input: any)\ - Checks if a date falls on today's calendar date.
- \is_weekday(d: any)\ - Check if a date falls on a weekday (Monday–Friday).
- \iso_day_name(d: any)\ - Return the English name of the day of the week.
- \millennium_of_date(d: any)\ - Return the millennium number of a date.
- \moon_phase(d: any)\ - Returns the approximate lunar phase for a given date.
- \next_month_same_day(d: any)\ - Return the same day number in the next month.
- \nth_weekday_of_month(year: integer, month: integer, weekday: integer, n: integer)\ - Return the date of the n-th occurrence of a weekday in a month.
- \ordinal_date_string(d: any)\ - Return the ISO ordinal date string for a date.
- \previous_month_same_day(d: any)\ - Return the same day number in the previous month.
- \quarter_end_date(year: integer, quarter: integer)\ - Return the last day of a given quarter.
- \quarter_start_date(year: integer, quarter: integer)\ - Return the first day of a given quarter.
- \semester_of_date(d: any)\ - Return the semester of the year for a given date.
- \semester_of_year(d: any)\ - Return the semester (1 or 2) of the year.
- \sidereal_time(d: any, longitude: number = 0.0)\ - Approximate local sidereal time in hours.
- \trimester_of_date(d: any)\ - Return the trimester number for a date.
- \week_parity(d: any)\ - Return whether the ISO week number is even or odd.
- \week_year(d: any)\ - Return the ISO week-numbering year for a date.
- \weeks_between_dates(date1: any, date2: any)\ - Calculate the number of complete weeks between two dates.
- \workdays_in_month(year: integer, month: integer)\ - Count the number of weekdays (Mon-Fri) in a given month.
- \zodiac_sign(d: any)\ - Returns the Western zodiac sign for a date.

## fxDate.date_operations

- \academic_year(d: any, start_month: integer = 9)\ - Returns the academic year label for a date.
- \add_days_from_now(days: integer)\ - Adds or subtracts a specified number of days from the current date and time.
- \add_microseconds(date_input: any, microseconds: integer)\ - Añade o resta un número específico de microsegundos a un objeto datetime.
- \add_months(date_input: any, months: integer)\ - Shifts a date forward or backward by a given number of months.
- \add_time_to_date(original_date: any, quantity: integer, unit: string)\ - Adds or subtracts a specified quantity of a given time unit to/from a date.
- \add_years(date_input: any, years: integer)\ - Shifts a date forward or backward by a given number of years.
- \age(birthdate: string)\ - Calculates the age in complete years from a birth date to today.
- \business_days_until(start: any, end: any, holidays: array = None)\ - Count working days between two dates, excluding weekends and holidays.
- \business_hours_between(start_dt: any, end_dt: any, work_start: any = 09:00:00, work_end: any = 17:00:00)\ - Calculates the number of business hours between two datetimes.
- \business_quarter_label(d: any)\ - Returns a business quarter label for a date.
- \calculate_days_between_dates(start_date: any, end_date: any)\ - Calculates the number of days between two dates.
- \clamp_date(d: any, min_date: any, max_date: any)\ - Clamp a date to the range [min_date, max_date].
- \countdown_days(target: any, from_date: any = None)\ - Counts the number of days until a target date.
- \cron_next_run(cron_expr: string, from_dt: any = None)\ - Returns the next datetime that matches a 5-field cron expression.
- \cron_previous_run(cron_expr: string, from_dt: any = None)\ - Returns the most recent past datetime matching a 5-field cron expression.
- \date_intervals(start_date: any, end_date: any, granularity: string, input_format: string = None)\ - Genera una lista de intervalos de fechas (inicio, fin) para una granularidad específica,
- \date_part(part: string, my_date: any, first_day_of_week: integer = 0, first_week_of_year: integer = 1)\ - Extrae una parte específica de una fecha y hora, similar a la función DatePart de VBA.
- \date_range(start: any, end: any, step: integer = 1, unit: string = days)\ - Generates a list of dates between *start* and *end* at regular intervals.
- \date_sequence(start: any, end: any, step_days: integer = 1)\ - Generates a list of dates from start to end (inclusive).
- \date_to_ordinal(d: any)\ - Return the proleptic Gregorian ordinal of a date.
- \date_to_week_label(d: any)\ - Returns an ISO week label for a date.
- \datedif(start_date: any, end_date: any, unit: string)\ - Calculates the difference between two dates in specified units.
- \dates_between(start_date: any, end_date: any)\ - Generates a list of dates between a start date and an end date, inclusive.
- \day_of_year(d: any)\ - Returns the ordinal day of the year (1–366).
- \daylight_hours(latitude: number, d: any)\ - Estimates hours of daylight for a given latitude and date.
- \days_360(start_date: any, end_date: any, method: string = us)\ - Calculates days between two dates based on a 360-day year (12 months of 30 days).
- \days_between(start_date: any, end_date: any, basis: integer = 0)\ - Calculates days between two dates based on a day-count basis.
- \days_in_month(year: integer, month: integer)\ - Returns the number of days in a given month and year.
- \days_remaining_in_year(d: any)\ - Returns the number of days remaining until 31 December.
- \decimal_hours_between(dt1: any, dt2: any)\ - Returns the decimal number of hours between two datetimes.
- \diff_time(start_date: any, end_date: any, unit: string)\ - Returns the whole-unit difference between two dates.
- \easter_date(year: integer)\ - Computes the date of Easter Sunday for a given year.
- \elapsed_business_days(start: any, end: any, holidays: array = None)\ - Counts business days (Mon–Fri) between two dates, excluding holidays.
- \end_of_month(date_input: any, input_format: string = None)\ - Returns a datetime object representing the last day of the month for a given date.
- \end_of_month_offset(date_input: any, months: integer)\ - Returns the last day of the month that is N months from date_input.
- \end_of_quarter(date_input: any, input_format: string = None)\ - Returns the last day of the quarter for a given date.
- \end_of_year(date_input: any, input_format: string = None)\ - Returns a datetime object representing the last day of the year for a given date.
- \filter_dates_by_weekday(dates: array, weekday: integer)\ - Filtra una lista de fechas, devolviendo solo aquellas que caen en un día específico de la semana.
- \first_day_of_week(date_input: any, week_start_day: integer = 0)\ - Calculates the first day of the week for a given date, allowing customization of the week's starting day.
- \fiscal_quarter(d: any, fiscal_start_month: integer = 1)\ - Returns the fiscal quarter (1-4) for a given date.
- \format_date_iso(d: any)\ - Format a date or datetime as ISO 8601 string (YYYY-MM-DD).
- \format_datetime_ampm(dt_object: any)\ - Formats a datetime object into a 12-hour (AM/PM) string representation.
- \format_datetime_iso(dt: any)\ - Format a datetime as ISO 8601 string (YYYY-MM-DDTHH:MM:SS).
- \full_months_between(start_date: any, end_date: any)\ - Devuelve el número de meses completos transcurridos entre start_date y end_date. No cuenta meses parciales.
- \full_years_between(start_date: any, end_date: any)\ - Devuelve el número de años completos transcurridos entre start_date y end_date. No cuenta años parciales.
- \generate_random_date(start_date: any = None, end_date: any = None, business_days_only: boolean = False)\ - Generates a random date within a specified range, optionally restricted to business days.
- \generate_random_datetime(start_dt: any = None, end_dt: any = None, tz_info: string = None)\ - Generates a random datetime object within a specified range, optionally localized to a timezone.
- \generate_random_time(min_time: any = None, max_time: any = None)\ - Generates a random time object within an optional specified range.
- \get_age_from_dob(dob: any, dob_format: string = None, as_of_date: any = None)\ - Calculates the age in full years from a date of birth.
- \get_date_component(date_input: any, component: any)\ - Extracts a specific component (day, month, year, hour, minute, or second) from a date.
- \get_first_business_day_of_month(year: integer, month: integer, holidays: array = None)\ - Devuelve el primer día hábil de un mes y año dados, considerando fines de semana y una lista opcional de festivos.
- \get_last_business_day_of_month(year: integer, month: integer, holidays: array = None)\ - Devuelve el último día hábil de un mes y año dados, considerando fines de semana y una lista opcional de festivos.
- \get_last_friday_of_month(year: integer, month: integer)\ - Calculates the date of the last Friday of a given month and year.
- \get_next_friday(date_input: any)\ - Calculates the date of the next upcoming Friday relative to a given date.
- \get_nth_weekday_of_month(year: integer, month: integer, weekday: integer, n: integer)\ - Calculates the date of the n-th occurrence of a specific weekday in a given month.
- \get_number_of_days_in_quarter(year: integer, quarter: integer)\ - Calculates the total number of days in a specific quarter of a given year.
- \get_number_of_days_in_year(year: integer)\ - Calculates the total number of days in a given year.
- \get_previous_friday(date_input: any)\ - Calculates the date of the immediately preceding Friday relative to a given date.
- \get_quarter(date_input: any, input_format: string = None)\ - Extrae el número de trimestre (1 a 4) al que pertenece una fecha dada.
- \get_season(date_input: any, hemisphere: string = northern, lang: string = en, format: string = None)\ - Dada una fecha, devuelve la estación del año en el idioma especificado para el hemisferio indicado.
- \get_week_of_year(date_input: any)\ - Returns the ISO 8601 week number for a given date.
- \get_week_range(year: integer, week_number: integer)\ - Retrieves the start and end dates (Monday to Sunday) for a specific ISO week of a given year.
- \get_working_days_in_range(start_date: any, end_date: any, holidays: array = None)\ - Calculates the number of working days within a given date range.
- \get_year_calendar_by_weeks(year: integer, start_date: any = None, end_date: any = None, granularity: string = week)\ - Generates a calendar table for a year, similar to DAX CALENDAR/CALENDARAUTO.
- \human_readable_duration(seconds: integer)\ - Format a number of seconds as a human-readable duration string.
- \intersection_of_date_ranges(start1: any, end1: any, start2: any, end2: any)\ - Si dos rangos de fechas se solapan, devuelve una tupla con las fechas de inicio y fin del período de intersección.
- \is_between_dates(target_date: any, start_date: any, end_date: any, inclusive: boolean = True, format: string = None)\ - Comprueba si una 'target_date' se encuentra entre 'start_date' y 'end_date'.
- \is_date_type(value: any)\ - Checks if an object is a date/time type (date, datetime, or time).
- \is_duration_greater_than(start_date: any, end_date: any, threshold_duration: any)\ - Comprueba si la duración entre start_date y end_date es estrictamente mayor que threshold_duration.
- \is_duration_less_than(start_date: any, end_date: any, threshold_duration: any)\ - Comprueba si la duración entre start_date y end_date es estrictamente menor que threshold_duration.
- \is_leap_year(year: integer)\ - Checks if a given year is a leap year.
- \is_same_business_day(date1: any, date2: any, holidays: array = None)\ - Check whether two dates fall on the same business day.
- \is_valid_date(date_input: any, date_format: string = %Y-%m-%d)\ - Validates if input is a valid date (datetime object or parseable string).
- \is_valid_time(hour: integer, minute: integer, second: integer = 0, microsecond: integer = 0)\ - Verifica si una combinación de hora, minuto, segundo y microsegundo dada representa una hora válida.
- \is_weekend(date_input: any, input_format: string = None, language: string = en)\ - Determines if a given date is a Saturday or a Sunday.
- \is_working_day(date_input: any, input_format: string = None, system: any = european)\ - Checks if a given date is a working day (Monday to Friday) based on the specified week numbering system.
- \iso_week_number(date_input: any)\ - Returns the ISO 8601 week number for a given date.
- \iso_week_start(year: integer, week: integer)\ - Return the Monday that starts a given ISO week.
- \last_day_of_month(date_input: any, input_format: string = None)\ - Returns the last day of the month for a given date.
- \last_day_of_week(date_input: any, week_start_day: integer = 0)\ - Calcula el último día de la semana para una fecha dada, permitiendo la personalización del día de inicio de la semana.
- \last_weekday_of_month(date_input: any, target_weekday: integer)\ - Returns the last occurrence of a specific weekday in the month of a given date.
- \midpoint_date(d1: any, d2: any)\ - Return the midpoint date between two dates.
- \month_name(date_input: any, language: string = en)\ - Gets the name of the month for a given date in the specified language.
- \months_between_dates(start_date: any, end_date: any)\ - Calcula el número de meses completos transcurridos entre dos fechas.
- \network_days_intl(start_date: any, end_date: any, weekend: integer = 1, holidays: array = None)\ - Calculates working days between two dates with custom weekend definition.
- \networkdays(start_date: any, end_date: any, holidays: array = None)\ - Counts working days (Mon-Fri) between two dates, excluding holidays.
- \networkdays_intl(start_date: any, end_date: any, weekend: integer = 1, holidays: array = None)\ - Counts working days between two dates with custom weekends.
- \next_business_day(d: any, holidays: array = None)\ - Returns the next business day after the given date.
- \next_full_moon(d: any)\ - Approximate date of the next full moon after *d*.
- \next_occurrence(weekday: integer, hour: integer = 0, minute: integer = 0, ref: any = None)\ - Find the next occurrence of a given weekday and time.
- \next_weekday(date_input: any, target_weekday: integer)\ - Returns the next occurrence of a specific weekday after a given date.
- \overlap_dates(start1: string, end1: string, start2: string, end2: string)\ - Checks whether two date ranges overlap.
- \overlap_days(start1: any, end1: any, start2: any, end2: any)\ - Counts the overlapping days between two date ranges.
- \parts_to_date(year: integer, month: integer, day: integer)\ - Create a date from year, month and day with overflow support.
- \parts_to_datetime(year: integer, month: integer, day: integer, hour: integer = 0, minute: integer = 0, second: integer = 0, microsecond: integer = 0)\ - Create a datetime from components with overflow support.
- \parts_to_time(hour: integer = 0, minute: integer = 0, second: integer = 0, microsecond: integer = 0)\ - Create a time object from hour, minute and second with overflow.
- \previous_business_day(d: any, holidays: array = None)\ - Returns the previous business day before the given date.
- \previous_weekday(date_input: any, target_weekday: integer)\ - Returns the most recent occurrence of a specific weekday before a given date.
- \quarters_between_dates(start_date: any, end_date: any)\ - Calculates the number of complete quarters between two dates.
- \recurring_dates(start: any, end: any, frequency: any = monthly, weekday: integer = None)\ - Generates recurring dates within a range.
- \relative_time(dt_input: any, reference: any = None, lang: string = en)\ - Describe a datetime relative to a reference point.
- \relative_time_description(dt_value: any, reference: any = None, language: string = en)\ - Returns a human-readable relative time string.
- \round_datetime(dt_value: any, unit: any = hour)\ - Rounds a datetime to the nearest unit.
- \semester(d: any)\ - Returns which semester (1 or 2) a date falls in.
- \set_date_component(date_input: any, kwargs: any)\ - Returns a new datetime object with specified components modified.
- \set_microseconds(date_input: any, microseconds: integer)\ - Devuelve un nuevo objeto datetime con los microsegundos establecidos al valor especificado.
- \shift_schedule(start_date: any, shift_days: integer, rest_days: integer, target_date: any)\ - Determines if a target date is a work or rest day in a cyclic rotation.
- \snap_to_weekday(date_input: any, target_weekday: integer, direction: string = next)\ - Snap a date to the nearest specified weekday.
- \start_of_month(date_input: any, input_format: string = None)\ - Returns a datetime object representing the first day of the month for a given date.
- \start_of_quarter(date_input: any, input_format: string = None)\ - Returns the first day of the quarter for a given date.
- \start_of_year(date_input: any, input_format: string = None)\ - Returns a datetime object representing the first day of the year for a given date.
- \sunrise_sunset(latitude: number, longitude: number, d: any)\ - Approximate sunrise and sunset times for a location (UTC).
- \time_difference(start_date: any, end_date: any, unit: string = days)\ - Calculates the difference between two dates in a specified unit of time.
- \time_from_datetime(datetime_object: any)\ - Extracts the time component from a datetime object.
- \time_overlap(start1: any, end1: any, start2: any, end2: any)\ - Return overlapping seconds between two time intervals.
- \time_zone_abbreviation(tz_name: string, ref: any = None)\ - Return the common abbreviation for a timezone at a given moment.
- \timedelta_to_components(timedelta_obj: any)\ - Convierte un objeto timedelta de Python en un diccionario con sus componentes.
- \trading_days_between(start: any, end: any, holidays: array = None, weekend: integer = (5, 6))\ - Count trading days between two dates.
- \week_number(date_input: any, system: integer = 1)\ - Returns the week number of the year for a given date.
- \week_of_month(date_input: any, start_of_week: integer = 0)\ - Devuelve el número de la semana del mes para una fecha dada.
- \weekday_name(date_input: any, language: string = en)\ - Gets the name of the day of the week for a given date in the specified language.
- \weekday_number(date_input: any, start_day: string = european)\ - Retrieves the day of the week as a number for a given date, supporting
- \workday(start_date: any, days: integer, holidays: array = None)\ - Returns the date after a given number of working days (Mon-Fri).
- \workday_intl(start_date: any, days: integer, weekend: integer = 1, holidays: array = None)\ - Returns the date after a given number of working days with custom weekends.
- \working_hours_in_month(year: integer, month: integer, hours_per_day: number = 8.0, holidays: array = None)\ - Calculate total working hours in a given month.
- \year_fraction(start_date: any, end_date: any, basis: integer = 0)\ - Calculates the fraction of year between two dates.

## fxDate.date_sys

- \current_date()\ - Gets the current local date (without the time component).
- \current_datetime()\ - Gets the current local date and time.
- \current_day()\ - Gets the current day of the month (1-31).
- \current_days_in_month()\ - Returns the number of days in the current month.
- \current_first_day_of_month()\ - Returns the first day of the current month at midnight.
- \current_is_leap_year()\ - Checks if the current year is a leap year.
- \current_is_weekend()\ - Checks if the current date is a weekend (Saturday or Sunday).
- \current_is_working_day()\ - Checks if the current date is a working day (Monday to Friday).
- \current_last_day_of_month()\ - Gets the last day of the current month.
- \current_last_friday_of_month()\ - Gets the last Friday of the current month.
- \current_month()\ - Gets the current month (1-12).
- \current_month_name(language: string = en)\ - Returns the name of the current month in the specified language.
- \current_next_friday()\ - Gets the next Friday after the current date.
- \current_previous_friday()\ - Gets the previous Friday before the current date.
- \current_quarter()\ - Returns the current quarter of the year (1-4).
- \current_season(hemisphere: string = northern, lang: string = en)\ - Returns the current season of the year.
- \current_time()\ - Gets the current local time (without the date component).
- \current_week_number()\ - Returns the current ISO week number (1-53).
- \current_weekday_name(language: string = en)\ - Gets the current weekday name for the current date.
- \current_weekday_number(start_day: string = european)\ - Retrieves the current day of the week as a number.
- \current_year()\ - Gets the current year.
- \get_local_now(tz: string = None)\ - Gets the current date and time in a specified timezone.
- \seconds_since_midnight()\ - Returns the number of seconds elapsed since midnight of the current day.

## fxExcel.database_formulas

- \DAVERAGE(database: array, field: string, criteria: object = None)\ - Calculate average of database field values matching criteria.
- \DCOUNT(database: array, field: string, criteria: object = None)\ - Count cells containing numbers in a database field that match criteria.
- \DCOUNTA(database: array, field: string, criteria: object = None)\ - Count non-empty cells in a database field that match criteria.
- \DGET(database: array, field: string, criteria: object)\ - Extract a single value from a database field that matches criteria.
- \DMAX(database: array, field: string, criteria: object = None)\ - Return maximum value from database field values matching criteria.
- \DMIN(database: array, field: string, criteria: object = None)\ - Return minimum value from database field values matching criteria.
- \DPRODUCT(database: array, field: string, criteria: object = None)\ - Multiply values from database field that match criteria.
- \DSTDEV(database: array, field: string, criteria: object = None)\ - Calculate sample standard deviation of database field values matching criteria.
- \DSTDEVP(database: array, field: string, criteria: object = None)\ - Calculate population standard deviation of database field values matching criteria.
- \DSUM(database: array, field: string, criteria: object = None)\ - Sum values from database field that match criteria.
- \DVAR(database: array, field: string, criteria: object = None)\ - Calculate sample variance of database field values matching criteria.
- \DVARP(database: array, field: string, criteria: object = None)\ - Calculate population variance of database field values matching criteria.

## fxExcel.date_formulas

- \DATE(year: integer, month: integer, day: integer)\ - Returns the Excel serial number corresponding to a particular date.
- \DATEDIF(start_date: any, end_date: any, unit: string)\ - Calculates the difference between dates in various units.
- \DATEVALUE(date_text: string)\ - Converts a date in text format to an Excel serial number.
- \DAY(serial_number: number)\ - Converts an Excel serial number to a day of the month.
- \DAYS(end_date: any, start_date: any)\ - Returns the number of days between two dates.
- \DAYS360(start_date: any, end_date: any, method: boolean = False)\ - Calculates days between two dates based on a 360-day year.
- \EDATE(start_date: any, months: integer)\ - Returns the serial number of a date months before or after.
- \EOMONTH(start_date: any, months: integer)\ - Returns the serial number of the last day of the month.
- \HOUR(serial_number: number)\ - Converts an Excel serial number to an hour value.
- \ISOWEEKNUM(date: any)\ - Returns the ISO week number.
- \MINUTE(serial_number: number)\ - Converts an Excel serial number to a minute value.
- \MONTH(serial_number: number)\ - Converts an Excel serial number to a month value.
- \NETWORKDAYS(start_date: any, end_date: any, holidays: array = None)\ - Returns the number of working days between two dates.
- \NETWORKDAYS_INTL(start_date: any, end_date: any, weekend: integer = 1, holidays: array = None)\ - Returns working days between two dates with custom weekend parameters.
- \NOW()\ - Returns the Excel serial number of the current date and time.
- \SECOND(serial_number: number)\ - Converts an Excel serial number to a second value.
- \TIME(hour: integer, minute: integer, second: integer)\ - Returns the Excel serial number for a particular time.
- \TIMEVALUE(time_text: string)\ - Converts a time in text format to a serial number.
- \TODAY()\ - Returns the Excel serial number of the current date.
- \WEEKDAY(serial_number: number, return_type: integer = 1)\ - Converts an Excel serial number to a day of the week.
- \WEEKNUM(serial_number: number, return_type: integer = 1)\ - Converts a serial number to a week number.
- \WORKDAY(start_date: number, days: integer, holidays: array = None)\ - Returns the date after a number of working days.
- \WORKDAY_INTL(start_date: number, days: integer, weekend: integer = 1, holidays: array = None)\ - Returns the date after working days with custom weekend.
- \YEAR(serial_number: number)\ - Converts an Excel serial number to a year value.
- \YEARFRAC(start_date: any, end_date: any, basis: integer = 0)\ - Returns the fraction of year between two dates.

## fxExcel.engineering_formulas

- \BESSELI(n: integer, x: integer)\ - Returns the modified Bessel function In(x).
- \BESSELJ(n: integer, x: integer)\ - Returns the Bessel function Jn(x).
- \BESSELK(n: integer, x: integer)\ - Returns the modified Bessel function Kn(x).
- \BESSELY(n: integer, x: integer)\ - Returns the Bessel function Yn(x).
- \BIN2DEC(binary: string)\ - Converts a binary string to decimal.
- \BIN2HEX(binary: string, places: integer = None)\ - Converts a binary string to hexadecimal.
- \BIN2OCT(binary: string, places: integer = None)\ - Converts a binary string to octal.
- \BITAND(number1: integer, number2: integer)\ - Performs a bitwise AND on two numbers.
- \BITLSHIFT(number: integer, shift: integer)\ - Performs a left shift on a number by shift bits.
- \BITOR(number1: integer, number2: integer)\ - Performs a bitwise OR on two numbers.
- \BITRSHIFT(number: integer, shift: integer)\ - Performs a right shift on a number by shift bits.
- \BITXOR(number1: integer, number2: integer)\ - Performs a bitwise XOR on two numbers.
- \COMPLEX(real: integer, imag: integer, suffix: string = i)\ - Creates a complex number from real and imaginary coefficients.
- \CONVERT(number: integer, from_unit: string, to_unit: string)\ - Converts a number from one measurement unit to another.
- \DEC2BIN(decimal: integer, places: integer = None)\ - Converts a decimal number to binary.
- \DEC2HEX(decimal: integer, places: integer = None)\ - Converts a decimal number to hexadecimal.
- \DEC2OCT(decimal: integer, places: integer = None)\ - Converts a decimal number to octal.
- \DELTA(value1: integer, value2: integer = 0)\ - Tests whether two values are equal. Returns 1 if equal, 0 otherwise.
- \ERF(x: integer)\ - Returns the error function erf(x).
- \ERFC(x: integer)\ - Returns the complementary error function erfc(x).
- \ERFC_PRECISE(x: integer)\ - Returns the complementary error function erfc(x) with precise calculation.
- \ERF_PRECISE(x: integer)\ - Returns the error function erf(x) with precise calculation.
- \GESTEP(number: integer, threshold: integer = 0)\ - Tests whether a number is greater than or equal to a threshold.
- \HEX2BIN(hexadecimal: string, places: integer = None)\ - Converts a hexadecimal string to binary.
- \HEX2DEC(hexadecimal: string)\ - Converts a hexadecimal string to decimal.
- \HEX2OCT(hexadecimal: string, places: integer = None)\ - Converts a hexadecimal string to octal.
- \IMABS(complex_num: any)\ - Returns the absolute value (modulus) of a complex number.
- \IMAGINARY(complex_num: any)\ - Returns the imaginary part of a complex number.
- \IMARGUMENT(complex_num: any)\ - Returns the argument (angle in radians) of a complex number.
- \IMCONJUGATE(complex_num: any)\ - Returns the complex conjugate of a complex number.
- \IMCOS(complex_num: any)\ - Returns the cosine of a complex number.
- \IMCOSH(complex_num: any)\ - Returns the hyperbolic cosine of a complex number.
- \IMCOT(complex_num: any)\ - Returns the cotangent of a complex number.
- \IMCSC(complex_num: any)\ - Returns the cosecant of a complex number.
- \IMCSCH(complex_num: any)\ - Returns the hyperbolic cosecant of a complex number.
- \IMDIV(complex_num1: any, complex_num2: any)\ - Returns the quotient of two complex numbers.
- \IMEXP(complex_num: any)\ - Returns the exponential of a complex number.
- \IMLN(complex_num: any)\ - Returns the natural logarithm of a complex number.
- \IMLOG10(complex_num: any)\ - Returns the common logarithm (base 10) of a complex number.
- \IMLOG2(complex_num: any)\ - Returns the base-2 logarithm of a complex number.
- \IMPOWER(complex_num: any, n: integer)\ - Returns a complex number raised to a power.
- \IMPRODUCT(complex_nums: array = ...)\ - Returns the product of complex numbers.
- \IMREAL(complex_num: any)\ - Returns the real part of a complex number.
- \IMSEC(complex_num: any)\ - Returns the secant of a complex number.
- \IMSECH(complex_num: any)\ - Returns the hyperbolic secant of a complex number.
- \IMSIN(complex_num: any)\ - Returns the sine of a complex number.
- \IMSINH(complex_num: any)\ - Returns the hyperbolic sine of a complex number.
- \IMSQRT(complex_num: any)\ - Returns the square root of a complex number.
- \IMSUB(complex_num1: any, complex_num2: any)\ - Returns the difference between two complex numbers.
- \IMSUM(complex_nums: array = ...)\ - Returns the sum of complex numbers.
- \IMTAN(complex_num: any)\ - Returns the tangent of a complex number.
- \OCT2BIN(octal: string, places: integer = None)\ - Converts an octal string to binary.
- \OCT2DEC(octal: string)\ - Converts an octal string to decimal.
- \OCT2HEX(octal: string, places: integer = None)\ - Converts an octal string to hexadecimal.

## fxExcel.financial_formulas

- \ACCRINT(issue: any, first_interest: any, settlement: any, rate: number, par: integer, frequency: integer, basis: integer = 0)\ - Return the accrued interest for a security that pays periodic interest.
- \ACCRINTM(issue: any, settlement: any, rate: number, par: integer, basis: integer = 0)\ - Return the accrued interest for a security that pays interest at maturity.
- \AMORDEGRC(cost: any, date_purchased: any, first_period: any, salvage: any, period: any, rate: any, basis: any = 0)\ - Calculate depreciation for each accounting period using a depreciation coefficient.
- \AMORLINC(cost: any, date_purchased: any, first_period: any, salvage: any, period: any, rate: any, basis: any = 0)\ - Calculate depreciation for each accounting period using a depreciation coefficient.
- \COUPDAYBS(settlement: any, maturity: any, frequency: integer)\ - Return the number of days from the beginning of the coupon period to the settlement date.
- \COUPDAYS(settlement: any, maturity: any, frequency: integer)\ - Return the number of days in the coupon period containing the settlement date.
- \COUPDAYSNC(settlement: any, maturity: any, frequency: integer)\ - Return the number of days from the settlement date to the next coupon date.
- \COUPNCD(settlement: any, maturity: any, frequency: integer)\ - Return the next coupon date after the settlement date.
- \COUPNUM(settlement: any, maturity: any, frequency: integer)\ - Return the number of coupons payable between settlement and maturity dates.
- \COUPPCD(settlement: any, maturity: any, frequency: integer)\ - Return the previous coupon date before the settlement date.
- \CUMIPMT(rate: number, nper: integer, pv: integer, start_period: integer, end_period: integer, type: integer)\ - Calculate cumulative interest paid between two periods.
- \CUMPRINC(rate: number, nper: integer, pv: integer, start_period: integer, end_period: integer, type: integer = 0)\ - Return the cumulative principal paid on a loan between two periods.
- \DB(cost: integer, salvage: integer, life: integer, period: integer, month: integer = 12)\ - Calculate depreciation using the fixed-declining balance method.
- \DDB(cost: integer, salvage: integer, life: integer, period: integer, factor: integer = 2)\ - Calculate depreciation using the double-declining balance method.
- \DISC(settlement: any, maturity: any, pr: any, redemption: any, basis: any = 0)\ - Calculate the discount rate for a security.
- \DOLLARDE(fractional_dollar: any, fraction: any)\ - Convert a dollar price expressed as a fraction into a decimal dollar price.
- \DOLLARFR(decimal_dollar: any, fraction: any)\ - Convert a decimal dollar price into a fractional dollar price.
- \DURATION(settlement: any, maturity: any, coupon: number, yld: number, frequency: integer, basis: integer = 0)\ - Return the Macaulay duration for an assumed par value of 100.
- \EFFECT(nominal_rate: number, npery: integer)\ - Calculate the effective annual interest rate.
- \FV(rate: number, nper: integer, pmt: integer, pv: integer = 0, type: integer = 0)\ - Calculate the future value of an investment.
- \FVSCHEDULE(principal: integer, schedule: array)\ - Calculate future value with a variable interest rate schedule.
- \INTRATE(settlement: any, maturity: any, investment: any, redemption: any, basis: any = 0)\ - Calculate the interest rate for a fully invested security.
- \IPMT(rate: number, per: integer, nper: integer, pv: integer, fv: integer = 0, type: integer = 0)\ - Calculate the interest payment for a given period.
- \IRR(values: array, guess: number = 0.1)\ - Calculate the internal rate of return for a series of cash flows.
- \ISPMT(rate: any, per: any, nper: any, pv: any)\ - Calculate the interest paid during a specific period of an investment.
- \MDURATION(settlement: any, maturity: any, coupon: number, yld: number, frequency: integer, basis: integer = 0)\ - Return the modified duration for an assumed par value of 100.
- \MIRR(values: array, finance_rate: number, reinvest_rate: number)\ - Calculate the modified internal rate of return.
- \NOMINAL(effect_rate: number, npery: integer)\ - Calculate the nominal annual interest rate.
- \NPER(rate: number, pmt: integer, pv: integer, fv: integer = 0, type: integer = 0)\ - Calculate the number of periods for an investment.
- \NPV(rate: number, values: array = ...)\ - Calculate the net present value of an investment.
- \ODDFPRICE(settlement: any, maturity: any, issue: any, first_coupon: any, rate: any, yld: any, redemption: any, frequency: any, basis: any = 0)\ - Calculate the price per $100 face value of a security with an odd first period.
- \ODDFYIELD(settlement: any, maturity: any, issue: any, first_coupon: any, rate: any, pr: any, redemption: any, frequency: any, basis: any = 0)\ - Calculate the yield of a security with an odd first period.
- \ODDLPRICE(settlement: any, maturity: any, last_interest: any, rate: any, yld: any, redemption: any, frequency: any, basis: any = 0)\ - Calculate the price per $100 face value of a security with an odd last period.
- \ODDLYIELD(settlement: any, maturity: any, last_interest: any, rate: any, pr: any, redemption: any, frequency: any, basis: any = 0)\ - Calculate the yield of a security with an odd last period.
- \PDURATION(rate: number, pv: integer, fv: integer)\ - Calculate the number of periods required for an investment to reach a specified value.
- \PMT(rate: number, nper: integer, pv: integer, fv: integer = 0, type: integer = 0)\ - Calculate the payment for a loan.
- \PPMT(rate: number, per: integer, nper: integer, pv: integer, fv: integer = 0, type: integer = 0)\ - Return the principal payment for a given period.
- \PRICE(settlement: any, maturity: any, rate: number, yld: number, redemption: integer, frequency: integer, basis: integer = 0)\ - Return the price per $100 face value of a security that pays periodic interest.
- \PRICEDISC(settlement: any, maturity: any, discount: number, redemption: integer, basis: integer = 0)\ - Return the price per $100 face value of a discounted security.
- \PRICEMAT(settlement: any, maturity: any, issue: any, rate: number, yld: number, basis: integer = 0)\ - Return the price per $100 face value of a security that pays interest at maturity.
- \PV(rate: number, nper: integer, pmt: integer, fv: integer = 0, type: integer = 0)\ - Calculate the present value of an investment.
- \RATE(nper: integer, pmt: integer, pv: integer, fv: integer = 0, type: integer = 0, guess: number = 0.1)\ - Calculate the interest rate per period of an annuity.
- \RECEIVED(settlement: any, maturity: any, investment: any, discount: any, basis: any = 0)\ - Calculate the amount received at maturity for a fully invested security.
- \RRI(nper: integer, pv: integer, fv: integer)\ - Calculate an equivalent interest rate for the growth of an investment.
- \SLN(cost: integer, salvage: integer, life: integer)\ - Calculate straight-line depreciation of an asset for one period.
- \SYD(cost: integer, salvage: integer, life: integer, per: integer)\ - Calculate sum-of-years digits depreciation.
- \TBILLEQ(settlement: any, maturity: any, discount: any)\ - Calculate the bond-equivalent yield for a Treasury bill.
- \TBILLPRICE(settlement: any, maturity: any, discount: any)\ - Calculate the price per $100 face value for a Treasury bill.
- \TBILLYIELD(settlement: any, maturity: any, pr: any)\ - Calculate the yield for a Treasury bill.
- \VDB(cost: integer, salvage: integer, life: integer, start_period: integer, end_period: integer, factor: integer = 2, no_switch: boolean = False)\ - Calculate depreciation using the double-declining balance or other method.
- \XIRR(values: array, dates: array, guess: number = 0.1)\ - Return the internal rate of return for a schedule of cash flows.
- \XNPV(rate: number, values: array, dates: array)\ - Return the net present value for a schedule of cash flows.
- \YIELD(settlement: any, maturity: any, rate: number, pr: integer, redemption: integer, frequency: integer, basis: integer = 0)\ - Return the yield of a security that pays periodic interest.
- \YIELDDISC(settlement: any, maturity: any, pr: integer, redemption: integer, basis: integer = 0)\ - Return the annual yield for a discounted security.
- \YIELDMAT(settlement: any, maturity: any, issue: any, rate: number, pr: integer, basis: integer = 0)\ - Return the annual yield of a security that pays interest at maturity.

## fxExcel.information_formulas

- \ERROR_TYPE(error_val: any)\ - Return a number corresponding to an error type.
- \ISBLANK(value: any)\ - Return TRUE if the value is blank.
- \ISERR(value: any)\ - Return TRUE if the value is any error value except #N/A.
- \ISERROR(value: any)\ - Return TRUE if the value is any error value.
- \ISEVEN(number: integer)\ - Return TRUE if the number is even.
- \ISLOGICAL(value: any)\ - Return TRUE if the value is a logical value.
- \ISNA(value: any)\ - Return TRUE if the value is the #N/A error value.
- \ISNONTEXT(value: any)\ - Return TRUE if the value is not text.
- \ISNUMBER(value: any)\ - Return TRUE if the value is a number.
- \ISODD(number: integer)\ - Return TRUE if the number is odd.
- \ISOMITTED(value: any)\ - Check if a value is missing in a LAMBDA expression.
- \ISTEXT(value: any)\ - Return TRUE if the value is text.
- \N(value: any)\ - Return a value converted to a number.
- \NA()\ - Return the error value #N/A.
- \TYPE(value: any)\ - Returns the type of a value as a numeric code.

## fxExcel.logic_formulas

- \AND(args: array = ...)\ - Return TRUE if all arguments are TRUE.
- \BYCOL(array: array, lambda_func: array)\ - Apply a LAMBDA to each column and return an array of results.
- \BYROW(array: array, lambda_func: array)\ - Apply a LAMBDA to each row and return an array of results.
- \FALSE()\ - Return the logical value FALSE.
- \IF(logical_test: any, value_if_true: any, value_if_false: any = None)\ - Perform a logical test and return one value for TRUE and another for FALSE.
- \IFERROR(value: any, value_if_error: any)\ - Return a value if an expression results in an error, otherwise return the expression result.
- \IFNA(value: any, value_if_na: any)\ - Return a value if the expression results in #N/A or None, otherwise return the expression result.
- \IFS(args: array = ...)\ - Check multiple conditions and return the value corresponding to the first TRUE condition.
- \LAMBDA(args: array = ..., expression: any = None)\ - Create a reusable lambda function with named parameters.
- \LET(kwargs: any)\ - Assign names to calculation results and evaluate a final expression.
- \MAKEARRAY(rows: integer, cols: integer, lambda_func: integer)\ - Return a calculated array of a specified row and column size by applying a LAMBDA.
- \MAP(args: array = ...)\ - Return an array formed by mapping each value in arrays to a new value by applying a LAMBDA.
- \NOT(value: any)\ - Reverse the logical value of its argument.
- \OR(args: array = ...)\ - Return TRUE if any argument is TRUE.
- \REDUCE(initial_value: any, array: array, lambda_func: any)\ - Reduce an array to an accumulated value by applying a LAMBDA to each value.
- \SCAN(initial_value: any, array: array, lambda_func: any)\ - Scan an array by applying a LAMBDA to each value, returning intermediate results.
- \SWITCH(expression: any, args: array = ...)\ - Evaluate an expression and return a value from a list based on matching values.
- \TRUE()\ - Return the logical value TRUE.
- \XOR(args: array = ...)\ - Return a logical exclusive OR of all arguments.

## fxExcel.lookup_formulas

- \ADDRESS(row_num: integer, column_num: integer, abs_num: integer = 1, a1: boolean = True, sheet_text: string = )\ - Build a cell-reference string from row/column numbers.
- \CHOOSE(index_num: integer, values: array = ...)\ - Choose a value from a list of values based on index.
- \CHOOSECOLS(array: array, col_nums: array = ...)\ - Return specified columns from an array.
- \CHOOSEROWS(array: array, row_nums: array = ...)\ - Return specified rows from an array.
- \COLUMN(reference: array = None, col_index: integer = 1)\ - Return the column number of a reference.
- \COLUMNS(array: array)\ - Return the number of columns in an array or reference.
- \DROP(array: array, rows: integer = 0, columns: integer = 0)\ - Exclude a specified number of rows or columns from start or end of array.
- \EXPAND(array: array, rows: integer = None, columns: integer = None, pad_with: any = )\ - Expand an array to the specified dimensions, padding with a value.
- \FILTER(array: array, include: array, if_empty: any = None)\ - Filter a range of data based on criteria.
- \HLOOKUP(lookup_value: any, table_array: array, row_index_num: integer, range_lookup: boolean = True)\ - Search in the top row of an array and return value from specified row.
- \HSTACK(arrays: array = ...)\ - Append arrays horizontally and in sequence to return a larger array.
- \INDEX(array: array, row_num: integer = 0, column_num: integer = 0)\ - Use an index to choose a value from a reference or array.
- \INDIRECT(ref_text: string, a1: boolean = True)\ - Parse a cell-reference string into (row, column) indices.
- \LOOKUP(lookup_value: any, lookup_vector: array, result_vector: array = None)\ - Look up values in a vector or array.
- \MATCH(lookup_value: any, lookup_array: array, match_type: integer = 1)\ - Return the relative position of an item in an array.
- \OFFSET(reference: array, rows: integer, cols: integer, height: integer = None, width: integer = None)\ - Return a sub-range from a 2-D array offset from a starting cell.
- \ROW(reference: array = None, row_index: integer = 1)\ - Return the row number of a reference.
- \ROWS(array: array)\ - Return the number of rows in an array or reference.
- \SORT(array: array, sort_index: integer = 1, sort_order: integer = 1, by_col: boolean = False)\ - Sort the contents of a range or array.
- \SORTBY(array: array, by_array1: array, sort_order1: integer = 1, args: array = ...)\ - Sort the contents of a range based on corresponding values in other ranges.
- \TAKE(array: array, rows: integer = None, columns: integer = None)\ - Return a specified number of contiguous rows or columns from start or end of array.
- \TOCOL(array: array, ignore: integer = 0, scan_by_column: boolean = False)\ - Return the array as a single column.
- \TOROW(array: array, ignore: integer = 0, scan_by_column: boolean = False)\ - Return the array as a single row.
- \TRANSPOSE(array: array)\ - Transpose a 2-D array (swap rows and columns).
- \TRIMRANGE(array: array)\ - Trim blank rows and columns from the edges of a range or array.
- \UNIQUE(array: array, by_col: boolean = False, exactly_once: boolean = False)\ - Return a list of unique values from a list or range.
- \VLOOKUP(lookup_value: any, table_array: array, col_index_num: integer, range_lookup: boolean = True)\ - Search in the first column of an array and return value from specified column.
- \VSTACK(arrays: array = ...)\ - Append arrays vertically and in sequence to return a larger array.
- \WRAPCOLS(vector: array, wrap_count: integer, pad_with: any = None)\ - Wrap the provided row or column of values by columns after specified number of elements.
- \WRAPROWS(vector: array, wrap_count: integer, pad_with: any = None)\ - Wrap the provided row or column of values by rows after specified number of elements.
- \XLOOKUP(lookup_value: any, lookup_array: array, return_array: array, if_not_found: any = #N/A, match_mode: integer = 0, search_mode: integer = 1)\ - Search a range or array and return corresponding item from second range/array.
- \XMATCH(lookup_value: any, lookup_array: array, match_mode: integer = 0, search_mode: integer = 1)\ - Return the relative position of an item in an array.

## fxExcel.math_formulas

- \ABS(number: number)\ - Returns the absolute value of a number.
- \ACOS(number: number)\ - Returns the arccosine of a number in radians.
- \ACOSH(number: number)\ - Returns the inverse hyperbolic cosine of a number.
- \ACOT(number: number)\ - Returns the arccotangent of a number in radians.
- \ACOTH(number: number)\ - Returns the inverse hyperbolic cotangent of a number.
- \AGGREGATE(data: array, operation: string = sum)\ - Returns an aggregate calculation from a list.
- \ARABIC(roman: string)\ - Converts a Roman numeral to an Arabic number.
- \ASIN(number: number)\ - Returns the arcsine of a number in radians.
- \ASINH(number: number)\ - Returns the inverse hyperbolic sine of a number.
- \ATAN(number: number)\ - Returns the arctangent of a number in radians.
- \ATAN2(x: number, y: number)\ - Returns the arctangent from x and y coordinates.
- \ATANH(number: number)\ - Returns the inverse hyperbolic tangent of a number.
- \BASE(number: integer, radix: integer, min_length: integer = 0)\ - Converts a number to text representation with a given base.
- \CEILING(number: number, significance: number = 1)\ - Rounds a number up to the nearest multiple of significance.
- \CEILING_MATH(number: number, significance: number = 1, mode: integer = 0)\ - Rounds a number up to nearest multiple with mode control.
- \CEILING_PRECISE(number: number, significance: number = 1)\ - Rounds a number up to nearest multiple (always away from zero).
- \COMBIN(n: integer, k: integer)\ - Returns the number of combinations for given items.
- \COMBINA(n: integer, k: integer)\ - Returns the number of combinations with repetitions.
- \COS(number: number)\ - Returns the cosine of an angle in radians.
- \COSH(number: number)\ - Returns the hyperbolic cosine of a number.
- \COT(number: number)\ - Returns the cotangent of an angle in radians.
- \COTH(number: number)\ - Returns the hyperbolic cotangent of a number.
- \CSC(number: number)\ - Returns the cosecant of an angle in radians.
- \CSCH(number: number)\ - Returns the hyperbolic cosecant of a number.
- \DECIMAL(text: string, radix: integer)\ - Converts text representation of number in given base to decimal.
- \DEGREES(angle: number)\ - Converts radians to degrees.
- \EVEN(number: number)\ - Rounds a number up to the nearest even integer.
- \EXP(number: number)\ - Returns e raised to the power of a number.
- \FACT(number: integer)\ - Returns the factorial of a number.
- \FACTDOUBLE(number: integer)\ - Returns the double factorial of a number.
- \FLOOR(number: number, significance: number = 1)\ - Rounds a number down to the nearest multiple of significance.
- \FLOOR_MATH(number: number, significance: number = 1, mode: integer = 0)\ - Rounds a number down to nearest multiple with mode control.
- \FLOOR_PRECISE(number: number, significance: number = 1)\ - Rounds a number down to nearest multiple (always toward zero).
- \GCD(numbers: array = ...)\ - Returns the greatest common divisor of integers.
- \INT(number: number)\ - Rounds a number down to the nearest integer.
- \ISO_CEILING(number: number, significance: number = 1)\ - Rounds a number up to nearest multiple (ISO standard).
- \LCM(numbers: array = ...)\ - Returns the least common multiple of integers.
- \LN(number: number)\ - Returns the natural logarithm of a number.
- \LOG(number: number, base: number = 10)\ - Returns the logarithm of a number to a specified base.
- \LOG10(number: number)\ - Returns the base-10 logarithm of a number.
- \MDETERM(matrix: array)\ - Returns the matrix determinant of an array.
- \MINVERSE(matrix: array)\ - Returns the inverse of a square matrix.
- \MMULT(matrix1: array, matrix2: array)\ - Returns the matrix product of two arrays.
- \MOD(number: number, divisor: number)\ - Returns the remainder from division.
- \MROUND(number: number, multiple: number)\ - Rounds a number to the nearest multiple.
- \MULTINOMIAL(numbers: array = ...)\ - Returns the multinomial of a set of numbers.
- \MUNIT(dimension: integer)\ - Returns the identity matrix for the specified dimension.
- \ODD(number: number)\ - Rounds a number up to the nearest odd integer.
- \PI()\ - Returns the value of pi (π).
- \POWER(number: number, power: number)\ - Returns the result of a number raised to a power.
- \PRODUCT(numbers: array = ...)\ - Returns the product of numbers.
- \QUOTIENT(numerator: number, denominator: number)\ - Returns the integer portion of a division.
- \RADIANS(angle: number)\ - Converts degrees to radians.
- \RAND()\ - Returns a random number between 0 and 1.
- \RANDARRAY(rows: integer = 1, columns: integer = 1, min_val: number = 0, max_val: number = 1, whole_number: boolean = False)\ - Returns an array of random numbers.
- \RANDBETWEEN(bottom: integer, top: integer)\ - Returns a random integer between two numbers.
- \ROMAN(number: integer, form: integer = 0)\ - Converts an Arabic number to Roman numeral as text.
- \ROUND(number: number, num_digits: integer = 0)\ - Rounds a number to a specified number of digits.
- \ROUNDDOWN(number: number, num_digits: integer = 0)\ - Rounds a number down (toward zero).
- \ROUNDUP(number: number, num_digits: integer = 0)\ - Rounds a number up (away from zero).
- \SEC(number: number)\ - Returns the secant of an angle in radians.
- \SECH(number: number)\ - Returns the hyperbolic secant of a number.
- \SEQUENCE(rows: integer, columns: integer = 1, start: number = 1, step: number = 1)\ - Generates a list of sequential numbers in an array.
- \SERIESSUM(x: number, n: integer, m: integer, coefficients: array)\ - Returns the sum of a power series.
- \SIGN(number: number)\ - Returns the sign of a number.
- \SIN(number: number)\ - Returns the sine of an angle in radians.
- \SINH(number: number)\ - Returns the hyperbolic sine of a number.
- \SQRT(number: number)\ - Returns the positive square root of a number.
- \SQRTPI(number: number)\ - Returns the square root of (number * pi).
- \SUBTOTAL(function_num: integer, values: array = ...)\ - Returns a subtotal in a list or database.
- \SUM(numbers: array = ...)\ - Returns the sum of numbers.
- \SUMIF(values: array, criteria: string)\ - Sums numbers that meet a specified criterion.
- \SUMIFS(sum_range: array, criteria_pairs: array = ...)\ - Sums values that meet multiple criteria.
- \SUMPRODUCT(arrays: array = ...)\ - Returns the sum of the products of corresponding array components.
- \SUMSQ(numbers: array = ...)\ - Returns the sum of the squares of the arguments.
- \SUMX2MY2(array_x: array, array_y: array)\ - Returns the sum of the difference of squares.
- \SUMX2PY2(array_x: array, array_y: array)\ - Returns the sum of the sum of squares.
- \SUMXMY2(array_x: array, array_y: array)\ - Returns the sum of squares of differences.
- \TAN(number: number)\ - Returns the tangent of an angle in radians.
- \TANH(number: number)\ - Returns the hyperbolic tangent of a number.
- \TRANSPOSE(array: array)\ - Transposes a matrix (swaps rows and columns).
- \TRUNC(number: number, num_digits: integer = 0)\ - Truncates a number to an integer by removing decimals.

## fxExcel.statistic_formulas

- \AVEDEV(values: array)\ - Returns the average of the absolute deviations of data points from their mean.
- \AVERAGE(values: array = ...)\ - Returns the average (arithmetic mean) of the arguments.
- \AVERAGEA(values: array = ...)\ - Returns the average of arguments, including numbers, text, and logical values.
- \AVERAGEIF(range_values: array, criteria: any, average_range: array = None)\ - Returns the average of cells that meet a criterion.
- \AVERAGEIFS(average_range: array, criteria_pairs: array = ...)\ - Returns the average of cells that meet multiple criteria.
- \BETADIST(x: number, alpha: number, beta: number, A: number = 0, B: number = 1)\ - Returns the beta cumulative distribution (backward compat alias).
- \BETAINV(probability: number, alpha: number, beta: number, A: number = 0, B: number = 1)\ - Returns the inverse of the beta cumulative distribution (backward compat alias).
- \BETA_DIST(x: number, alpha: number, beta: number, cumulative: boolean = True, A: number = 0, B: number = 1)\ - Returns the beta probability distribution function.
- \BETA_INV(probability: number, alpha: number, beta: number, A: number = 0, B: number = 1)\ - Returns the inverse of the beta cumulative distribution function.
- \BINOMDIST(number_s: integer, trials: integer, probability_s: number, cumulative: boolean = True)\ - Returns the binomial distribution probability (backward compat alias).
- \BINOM_DIST(number_s: integer, trials: integer, probability_s: number, cumulative: boolean)\ - Returns the individual term binomial distribution probability.
- \BINOM_DIST_RANGE(trials: integer, probability_s: number, number_s: integer, number_s2: integer = None)\ - Returns the probability of a trial result using a binomial distribution.
- \BINOM_INV(trials: integer, probability_s: number, alpha: number)\ - Returns the smallest value for which the cumulative binomial distribution is >= criterion.
- \CHIDIST(x: number, deg_freedom: integer)\ - Returns the right-tailed chi-squared distribution (backward compat alias).
- \CHIINV(probability: number, deg_freedom: integer)\ - Returns the inverse of the right-tailed chi-squared distribution (backward compat alias).
- \CHISQ_DIST(x: number, deg_freedom: integer, cumulative: boolean = True)\ - Returns the chi-squared distribution.
- \CHISQ_DIST_RT(x: number, deg_freedom: integer)\ - Returns the right-tailed probability of the chi-squared distribution.
- \CHISQ_INV(probability: number, deg_freedom: integer)\ - Returns the inverse of the left-tailed probability of the chi-squared distribution.
- \CHISQ_INV_RT(probability: number, deg_freedom: integer)\ - Returns the inverse of the right-tailed probability of the chi-squared distribution.
- \CHISQ_TEST(actual_range: array, expected_range: array)\ - Returns the test for independence (chi-squared test).
- \CHITEST(actual_range: array, expected_range: array)\ - Returns the chi-squared test for independence (backward compat alias).
- \CONFIDENCE(alpha: number, standard_dev: number, size: integer)\ - Returns the confidence interval (backward compat alias).
- \CONFIDENCE_NORM(alpha: number, standard_dev: number, size: integer)\ - Returns the confidence interval for a population mean (normal distribution).
- \CONFIDENCE_T(alpha: number, standard_dev: number, size: integer)\ - Returns the confidence interval for a population mean (t-distribution).
- \CORREL(array1: array, array2: array)\ - Returns the correlation coefficient between two data sets.
- \COUNT(values: array = ...)\ - Counts the number of cells that contain numbers.
- \COUNTA(values: array = ...)\ - Counts the number of cells that are not empty.
- \COUNTBLANK(range_values: array)\ - Counts empty cells in a range.
- \COUNTIF(range_values: array, criteria: any)\ - Counts the number of cells that meet a criterion.
- \COUNTIFS(args: array = ...)\ - Counts cells that meet multiple criteria.
- \COVAR(array1: array, array2: array)\ - Returns covariance (legacy Excel 2007 function, equivalent to COVARIANCE.P).
- \COVARIANCE_P(array1: array, array2: array)\ - Returns population covariance, the average of the products of deviations.
- \COVARIANCE_S(array1: array, array2: array)\ - Returns sample covariance.
- \CRITBINOM(trials: integer, probability_s: number, alpha: number)\ - Returns the smallest value for cumulative binomial distribution (backward compat alias).
- \DEVSQ(values: array)\ - Returns the sum of squares of deviations.
- \EXPONDIST(x: number, lambda_: number, cumulative: boolean = True)\ - Returns the exponential distribution (backward compat alias).
- \EXPON_DIST(x: number, lambda_: number, cumulative: boolean)\ - Returns the exponential distribution.
- \FDIST(x: number, deg_freedom1: integer, deg_freedom2: integer)\ - Returns the right-tailed F probability distribution (backward compat alias).
- \FINV(probability: number, deg_freedom1: integer, deg_freedom2: integer)\ - Returns the inverse of the right-tailed F distribution (backward compat alias).
- \FISHER(x: number)\ - Returns the Fisher transformation.
- \FISHERINV(y: number)\ - Returns the inverse of the Fisher transformation.
- \FORECAST(x: number, known_y: array, known_x: array = None)\ - Returns a single predicted value along a linear trend (backward compat).
- \FORECAST_ETS(target_date: number, values: array, timeline: array, seasonality: integer = None, data_completion: integer = 1, aggregation: integer = 1)\ - Returns a forecasted value based on exponential smoothing (simplified).
- \FORECAST_ETS_CONFINT(target_date: number, values: array, timeline: array, confidence_level: number = 0.95, seasonality: integer = None, data_completion: integer = 1, aggregation: integer = 1)\ - Returns confidence interval for forecast (simplified).
- \FORECAST_ETS_SEASONALITY(values: array, timeline: array, data_completion: integer = 1, aggregation: integer = 1)\ - Returns the detected seasonality length (simplified).
- \FORECAST_ETS_STAT(values: array, timeline: array, statistic_type: integer, seasonality: integer = None, data_completion: integer = 1, aggregation: integer = 1)\ - Returns statistical value for ETS forecast (simplified).
- \FORECAST_LINEAR(x: number, known_y: array, known_x: array = None)\ - Returns a value along a linear trend using linear regression.
- \FREQUENCY(data_array: array, bins_array: array)\ - Returns a frequency distribution as a vertical array.
- \FTEST(array1: array, array2: array)\ - Returns the F-test result (backward compat alias).
- \F_DIST(x: number, deg_freedom1: integer, deg_freedom2: integer, cumulative: boolean = True)\ - Returns the F probability distribution.
- \F_DIST_RT(x: number, deg_freedom1: integer, deg_freedom2: integer)\ - Returns the right-tailed F probability distribution.
- \F_INV(probability: number, deg_freedom1: integer, deg_freedom2: integer)\ - Returns the inverse of the F probability distribution.
- \F_INV_RT(probability: number, deg_freedom1: integer, deg_freedom2: integer)\ - Returns the inverse of the right-tailed F probability distribution.
- \F_TEST(array1: array, array2: array)\ - Returns the result of an F-test.
- \GAMMA(number: number)\ - Returns the gamma function value.
- \GAMMADIST(x: number, alpha: number, beta: number, cumulative: boolean = True)\ - Returns the gamma distribution (backward compat alias).
- \GAMMAINV(probability: number, alpha: number, beta: number)\ - Returns the inverse of the gamma cumulative distribution (backward compat alias).
- \GAMMALN(x: number)\ - Returns the natural logarithm of the gamma function.
- \GAMMALN_PRECISE(x: number)\ - Returns the natural logarithm of the gamma function (precise version).
- \GAMMA_DIST(x: number, alpha: number, beta: number, cumulative: boolean)\ - Returns the gamma distribution.
- \GAMMA_INV(probability: number, alpha: number, beta: number)\ - Returns the inverse of the gamma cumulative distribution.
- \GAUSS(z: number)\ - Returns 0.5 less than the standard normal cumulative distribution.
- \GEOMEAN(values: array)\ - Returns the geometric mean of an array of positive numbers.
- \GROWTH(known_y: array, known_x: array = None, new_x: array = None, const: boolean = True)\ - Returns values along an exponential trend.
- \HARMEAN(values: array)\ - Returns the harmonic mean of a data set.
- \HYPGEOMDIST(sample_s: integer, number_sample: integer, population_s: integer, number_pop: integer)\ - Returns the hypergeometric distribution (backward compat alias).
- \HYPGEOM_DIST(sample_s: integer, number_sample: integer, population_s: integer, number_pop: integer, cumulative: boolean)\ - Returns the hypergeometric distribution.
- \INTERCEPT(known_y: array, known_x: array = None)\ - Returns the intercept of the linear regression line.
- \KURT(values: array)\ - Returns the kurtosis of a data set (excess kurtosis).
- \LARGE(array: array, k: integer)\ - Returns the k-th largest value in a data set.
- \LINEST(known_y: array, known_x: array = None, const: boolean = True, stats_flag: boolean = False)\ - Returns statistics for a linear trend.
- \LOGEST(known_y: array, known_x: array = None, const: boolean = True, stats_flag: boolean = False)\ - Returns parameters of an exponential trend.
- \LOGINV(probability: number, mean: number, standard_dev: number)\ - Returns the inverse of the lognormal cumulative distribution (backward compat alias).
- \LOGNORMDIST(x: number, mean: number, standard_dev: number)\ - Returns the cumulative lognormal distribution (backward compat alias).
- \LOGNORM_DIST(x: number, mean: number, standard_dev: number, cumulative: boolean)\ - Returns the lognormal distribution.
- \LOGNORM_INV(probability: number, mean: number, standard_dev: number)\ - Returns the inverse of the lognormal cumulative distribution.
- \MAX(values: array = ...)\ - Returns the largest value in a set of values.
- \MAXA(values: array = ...)\ - Returns the largest value in a set of values (includes text and logical values).
- \MAXIFS(max_range: array, args: array = ...)\ - Returns the maximum value among cells specified by a given set of conditions.
- \MEDIAN(values: array = ...)\ - Returns the median (middle value) of the given numbers.
- \MIN(values: array = ...)\ - Returns the minimum value from a set of values.
- \MINA(values: array = ...)\ - Returns the minimum value, including numbers, text, and logical values.
- \MINIFS(min_range: array, criteria_pairs: array = ...)\ - Returns the minimum value among cells specified by a set of conditions.
- \MODE(values: array)\ - Returns the most common value in a data set (backward compat alias).
- \MODE_MULT(values: array)\ - Returns a vertical array of the most frequently occurring values in a range.
- \MODE_SNGL(values: array)\ - Returns the most common value in a data set.
- \NEGBINOMDIST(number_f: integer, number_s: integer, probability_s: number)\ - Returns the negative binomial distribution (backward compat alias).
- \NEGBINOM_DIST(number_f: integer, number_s: integer, probability_s: number, cumulative: boolean = False)\ - Returns the negative binomial distribution.
- \NORMDIST(x: number, mean: number, standard_dev: number, cumulative: boolean = True)\ - Returns the normal cumulative distribution (backward compat alias).
- \NORMINV(probability: number, mean: number, standard_dev: number)\ - Returns the inverse of the normal cumulative distribution (backward compat alias).
- \NORMSDIST(z: number)\ - Returns the standard normal cumulative distribution (backward compat alias).
- \NORMSINV(probability: number)\ - Returns the inverse of the standard normal cumulative distribution (backward compat alias).
- \NORM_DIST(x: number, mean: number, standard_dev: number, cumulative: boolean = False)\ - Returns the normal cumulative distribution.
- \NORM_INV(probability: number, mean: number, standard_dev: number)\ - Returns the inverse of the normal cumulative distribution.
- \NORM_S_DIST(z: number, cumulative: boolean = False)\ - Returns the standard normal cumulative distribution.
- \NORM_S_INV(probability: number)\ - Returns the inverse of the standard normal cumulative distribution.
- \PEARSON(array1: array, array2: array)\ - Returns the Pearson product-moment correlation coefficient.
- \PERCENTILE(array: array, k: number)\ - Returns the k-th percentile (backward compat alias).
- \PERCENTILE_EXC(array: array, k: number)\ - Returns the k-th percentile of values (k in range 0..1, exclusive).
- \PERCENTILE_INC(array: array, k: number)\ - Returns the k-th percentile of values in a range.
- \PERCENTRANK(array: array, x: number, significance: integer = 3)\ - Returns the percentage rank (backward compat alias).
- \PERCENTRANK_EXC(array: array, x: number, significance: integer = 3)\ - Returns the rank of a value as a percentage (0..1, exclusive).
- \PERCENTRANK_INC(array: array, x: number, significance: integer = 3)\ - Returns the percentage rank of a value in a data set.
- \PERMUT(number: integer, number_chosen: integer)\ - Returns the number of permutations for a given number of objects.
- \PERMUTATIONA(number: integer, number_chosen: integer)\ - Returns the number of permutations with repetition.
- \PHI(x: number)\ - Returns the value of the density function for a standard normal distribution.
- \POISSON(x: integer, mean: number, cumulative: boolean = True)\ - Returns the Poisson distribution (backward compat alias).
- \POISSON_DIST(x: integer, mean: number, cumulative: boolean = False)\ - Returns the Poisson distribution.
- \PROB(x_range: array, prob_range: array, lower_limit: number, upper_limit: number = None)\ - Returns the probability that values fall within a range.
- \QUARTILE(array: array, quart: integer)\ - Returns the quartile of a data set (backward compat alias).
- \QUARTILE_EXC(array: array, quart: integer)\ - Returns the quartile of a data set (exclusive of 0 and 4).
- \QUARTILE_INC(array: array, quart: integer)\ - Returns the quartile of a data set (inclusive of 0 and 4).
- \RANK(number: number, ref: array, order: integer = 0)\ - Returns the rank of a number in a list (backward compatibility alias for RANK.EQ).
- \RANK_AVG(number: number, ref: array, order: integer = 0)\ - Returns the rank of a number in a list, averaging for ties.
- \RANK_EQ(number: number, ref: array, order: integer = 0)\ - Returns the rank of a number in a list. Duplicate values receive the same rank.
- \RSQ(known_y: array, known_x: array)\ - Returns the R-squared value of the linear regression line.
- \SKEW(values: array = ...)\ - Returns the skewness of a distribution.
- \SKEW_P(values: array = ...)\ - Returns the skewness of a distribution based on a population.
- \SLOPE(known_y: array, known_x: array = None)\ - Returns the slope of the linear regression line.
- \SMALL(array: array, k: integer)\ - Returns the k-th smallest value in a data set.
- \STANDARDIZE(x: number, mean: number, standard_dev: number)\ - Returns a normalized value from a distribution.
- \STDEV(values: array = ...)\ - Calculates standard deviation based on a sample (backward compat alias).
- \STDEVA(values: array = ...)\ - Estimates standard deviation based on a sample, including text and logical values.
- \STDEVP(values: array = ...)\ - Returns population standard deviation (backward compat alias).
- \STDEVPA(values: array = ...)\ - Calculates standard deviation based on population, including text and logical values.
- \STDEV_P(values: array = ...)\ - Calculates standard deviation based on the entire population.
- \STDEV_S(values: array = ...)\ - Estimates standard deviation based on a sample.
- \STEYX(known_y: array, known_x: array)\ - Returns the standard error of the predicted y-value for each x in the regression.
- \TDIST(x: number, deg_freedom: integer, tails: integer = 2)\ - Returns the Student's t-distribution (backward compat alias).
- \TINV(probability: number, deg_freedom: integer)\ - Returns the inverse of the two-tailed Student's t-distribution (backward compat alias).
- \TREND(known_y: array, known_x: array = None, new_x: array = None, const: boolean = True)\ - Returns values along a linear trend using least-squares regression.
- \TRIMMEAN(array: array, fraction: number)\ - Returns the mean of the interior of a data set.
- \TTEST(array1: array, array2: array, tails: integer = 2, test_type: integer = 2)\ - Returns the probability from a t-test (backward compat alias).
- \T_DIST(x: number, deg_freedom: integer, cumulative: boolean = False)\ - Returns the Student's t-distribution.
- \T_DIST_2T(x: number, deg_freedom: integer)\ - Returns the two-tailed Student's t-distribution.
- \T_DIST_RT(x: number, deg_freedom: integer)\ - Returns the right-tailed Student's t-distribution.
- \T_INV(probability: number, deg_freedom: integer)\ - Returns the left-tailed inverse of the Student's t-distribution.
- \T_INV_2T(probability: number, deg_freedom: integer)\ - Returns the two-tailed inverse of the Student's t-distribution.
- \T_TEST(array1: array, array2: array, tails: integer = 2, test_type: integer = 1)\ - Returns the probability associated with a Student's t-Test.
- \VAR(values: array = ...)\ - Calculates variance based on a sample (backward compat alias).
- \VARA(values: array = ...)\ - Estimates variance based on a sample, including text and logical values.
- \VARP(values: array = ...)\ - Returns population variance (backward compat alias).
- \VARPA(values: array = ...)\ - Calculates variance based on population, including text and logical values.
- \VAR_P(values: array = ...)\ - Calculates variance based on the entire population.
- \VAR_S(values: array = ...)\ - Estimates variance based on a sample.
- \WEIBULL(x: number, alpha: number, beta: number, cumulative: boolean = True)\ - Returns the Weibull distribution (backward compat alias).
- \WEIBULL_DIST(x: number, alpha: number, beta: number, cumulative: boolean = False)\ - Returns the Weibull distribution.
- \ZTEST(array: array, x: number, sigma: number = None)\ - Returns the one-tailed probability-value of a z-test (backward compat alias).
- \Z_TEST(array: array, x: number, sigma: number = None)\ - Returns the one-tailed P-value of a z-test.

## fxExcel.text_formulas

- \ARRAYTOTEXT(array: any, format_type: integer = 0)\ - Returns a text representation of an array of values.
- \ASC(text: string)\ - Converts full-width (double-byte) characters to half-width (single-byte).
- \BAHTTEXT(number: integer)\ - Converts a number to text using Thai Baht currency format.
- \CHAR(number: integer)\ - Returns the character specified by the code number.
- \CLEAN(text: string)\ - Removes all nonprintable characters from text.
- \CODE(text: string)\ - Returns a numeric code for the first character in a text string.
- \CONCAT(args: array = ...)\ - Combines text from multiple ranges and/or strings.
- \CONCATENATE(args: array = ...)\ - Joins several text strings into one text string.
- \DBCS(text: string)\ - Converts half-width (single-byte) characters to full-width (double-byte).
- \DOLLAR(number: integer, decimals: integer = 2)\ - Converts a number to text using currency format ($).
- \EXACT(text1: string, text2: string)\ - Checks whether two text strings are exactly the same.
- \FIND(find_text: string, within_text: string, start_num: integer = 1)\ - Finds one text string within another (case-sensitive).
- \FINDB(find_bytes: string, within_text: string, start_byte: integer = 1)\ - Finds one text value within another based on bytes (case-sensitive).
- \FIXED(number: integer, decimals: integer = 2, no_commas: boolean = False)\ - Formats a number as text with a fixed number of decimals.
- \LEFT(text: string, num_chars: integer = 1)\ - Returns the leftmost characters from a text string.
- \LEFTB(text: string, num_bytes: integer = 1)\ - Returns the leftmost characters based on number of bytes.
- \LEN(text: string)\ - Returns the number of characters in a text string.
- \LENB(text: string)\ - Returns the number of bytes in text.
- \LOWER(text: string)\ - Converts text to lowercase.
- \MID(text: string, start_num: integer, num_chars: integer)\ - Returns characters from the middle of a text string.
- \MIDB(text: string, start_byte: integer, num_bytes: integer)\ - Returns characters from the middle based on bytes.
- \NUMBERVALUE(text: string, decimal_separator: string = ., group_separator: string = ,)\ - Converts text to number in a locale-independent manner.
- \PROPER(text: string)\ - Capitalizes the first letter in each word of a text string.
- \REGEXEXTRACT(text: string, pattern: string, match_mode: integer = 0, capture_group: integer = 0)\ - Extracts strings within text that match the regex pattern.
- \REGEXREPLACE(text: string, pattern: string, replacement: string, match_mode: integer = 0)\ - Replaces strings within text that match the regex pattern with replacement.
- \REGEXTEST(text: string, pattern: string, match_mode: integer = 0)\ - Tests whether any part of the text matches the regex pattern.
- \REPLACE(old_text: string, start_num: integer, num_chars: integer, new_text: string)\ - Replaces part of a text string with a different text string.
- \REPLACEB(old_text: string, start_byte: integer, num_bytes: integer, new_text: string)\ - Replaces part of a text string based on bytes.
- \REPT(text: string, number_times: integer)\ - Repeats text a given number of times.
- \RIGHT(text: string, num_chars: integer = 1)\ - Returns the rightmost characters from a text string.
- \RIGHTB(text: string, num_bytes: integer = 1)\ - Returns the rightmost characters based on number of bytes.
- \SEARCH(find_text: string, within_text: string, start_num: integer = 1)\ - Finds one text string within another (case-insensitive).
- \SEARCHB(find_bytes: string, within_text: string, start_byte: integer = 1)\ - Finds one text value within another based on bytes (case-insensitive).
- \SUBSTITUTE(text: string, old_text: string, new_text: string, instance_num: integer = None)\ - Substitutes new text for old text in a text string.
- \T(value: any)\ - Converts its argument to text.
- \TEXT(value: integer, format_text: string)\ - Formats a number and converts it to text.
- \TEXTAFTER(text: string, delimiter: string, instance_num: integer = 1, match_mode: integer = 0, match_end: integer = 0, if_not_found: any = None)\ - Returns text that occurs after a given character or substring.
- \TEXTBEFORE(text: string, delimiter: string, instance_num: integer = 1, match_mode: integer = 0, match_end: integer = 0, if_not_found: any = None)\ - Returns text that occurs before a given character or substring.
- \TEXTJOIN(delimiter: string, ignore_empty: boolean, args: array = ...)\ - Combines text from multiple ranges with a delimiter.
- \TEXTSPLIT(text: string, col_delimiter: string =  , row_delimiter: string = None, ignore_empty: boolean = False, match_mode: integer = 0, pad_with: any = None)\ - Splits text into rows and columns using delimiters.
- \TRIM(text: string)\ - Removes all spaces from text except single spaces between words.
- \UNICHAR(number: integer)\ - Returns the Unicode character referenced by the given numeric value.
- \UNICODE(text: string)\ - Returns the number (code point) corresponding to the first character of the text.
- \UPPER(text: string)\ - Converts text to uppercase.
- \VALUE(text: string)\ - Converts text that represents a number to a number.
- \VALUETOTEXT(value: any, format_type: integer = 0)\ - Returns text from any specified value.

## fxNumeric.arithmetic_functions

- \absolute_value(x: integer)\ - Returns the absolute (non-negative) value of a number.
- \adaptive_simpson(f: number, a: number, b: number, tol: number = 1e-10, max_depth: integer = 50)\ - Adaptive Simpson's rule for numerical integration.
- \additive_persistence(n: integer)\ - Count iterations to reduce n to a single digit by summing its digits.
- \adjugate_matrix(matrix: array)\ - Computes the adjugate (classical adjoint) of a square matrix.
- \bessel_i(x: number, n: integer)\ - Modified Bessel function of the first kind, order n.
- \bessel_j(x: number, n: integer)\ - Bessel function of the first kind, order n.
- \bessel_k(x: number, n: integer)\ - Modified Bessel function of the second kind, order n.
- \bessel_y(x: number, n: integer)\ - Bessel function of the second kind, order n.
- \bisection_method(f: number, a: number, b: number, tol: number = 1e-10, max_iter: integer = 100)\ - Finds a root of f(x) = 0 in [a, b] using the bisection method.
- \bit_count(number: integer)\ - Counts the number of set bits (1-bits) in a non-negative integer.
- \bit_lshift(number: integer, shift_amount: integer)\ - Shift number left by shift_amount bits.
- \bit_rshift(number: integer, shift_amount: integer)\ - Shift number right by shift_amount bits.
- \bitwise_and(a: integer, b: integer)\ - Returns the bitwise AND of two non-negative integers.
- \bitwise_not(number: integer, bit_width: integer = 32)\ - Returns the bitwise NOT of a non-negative integer within a given bit width.
- \bitwise_or(a: integer, b: integer)\ - Returns the bitwise OR of two non-negative integers.
- \bitwise_xor(a: integer, b: integer)\ - Returns the bitwise XOR of two non-negative integers.
- \cantor_pairing(a: integer, b: integer)\ - Compute the Cantor pairing function for two non-negative integers.
- \cantor_unpairing(z: integer)\ - Invert the Cantor pairing function.
- \chebyshev_nodes(n: integer, a: number = -1.0, b: number = 1.0)\ - Generate *n* Chebyshev nodes on the interval [a, b].
- \cholesky_decomposition(matrix: array)\ - Performs Cholesky decomposition of a symmetric positive-definite matrix.
- \collatz_sequence(n: integer)\ - Generate the Collatz sequence starting from n.
- \collatz_steps(n: integer)\ - Count the number of Collatz steps to reach 1.
- \combinations(n: integer, k: integer)\ - Calculates the number of combinations C(n, k).
- \combinations_with_repetition(n: integer, k: integer)\ - Calculates combinations with repetition (multiset coefficient).
- \common_log(x: number)\ - Calculates the common logarithm (base 10) of a number.
- \complementary_error_function(x: number)\ - Calculates the complementary error function erfc(x) = 1 - erf(x).
- \continued_fraction(numerator: integer, denominator: integer, max_terms: integer = 20)\ - Computes the continued fraction representation of a rational number.
- \cross_product(vec_a: array, vec_b: array)\ - Computes the cross product of two 3D vectors.
- \cube_root(x: integer)\ - Calculates the cube root of a number.
- \delta(number_1: number, number_2: number = 0)\ - Tests whether two values are equal (Kronecker delta).
- \digit_product(n: integer)\ - Compute the product of digits of a positive integer.
- \digital_sum(n: integer)\ - Compute the digit sum of an integer.
- \dot_product(vec_a: array, vec_b: array)\ - Computes the dot product of two vectors.
- \double_factorial(n: integer)\ - Calculates the double factorial of a non-negative integer.
- \error_function(x: number)\ - Calculates the error function erf(x).
- \euler_method(f: any, y0: number, t0: number, t_end: number, steps: integer = 100)\ - Solve an ODE y' = f(t, y) using Euler's method.
- \exp(x: number)\ - Calculates e raised to the power of x.
- \expm1(x: number)\ - Calculates e^x - 1 with high precision for small x.
- \factorial(n: integer)\ - Calculates the factorial of a non-negative integer.
- \fibonacci(n: integer)\ - Returns the n-th Fibonacci number (0-indexed).
- \fixed_point_iteration(g: number, x0: number, tol: number = 1e-10, max_iter: integer = 100)\ - Find a fixed point of *g*, i.e. x such that g(x) = x.
- \floor_division(numerator: integer, denominator: integer)\ - Performs integer (floor) division, truncating the result downwards.
- \force_float_division(numerator: integer, denominator: integer)\ - Performs division ensuring the result is always a float.
- \gamma(x: number)\ - Calculates the Gamma function Γ(x).
- \gaussian_quadrature(f: any, a: number, b: number, n: integer = 5)\ - Numerical integration via Gauss-Legendre quadrature.
- \gcd_list(numbers: array)\ - Calculates the GCD of a list of integers.
- \gestep(number: number, step: number = 0)\ - Tests whether a number is greater than or equal to a step value.
- \greatest_common_divisor(a: integer, b: integer)\ - Calculates the greatest common divisor (GCD) of two integers.
- \identity_matrix(size: integer)\ - Creates an identity matrix of the given size.
- \integer_partition_distinct(n: integer)\ - Count partitions of n into distinct parts using DP.
- \is_automorphic(n: integer)\ - Check if n is an automorphic number (n² ends with n).
- \is_happy_number(n: integer)\ - Check if n is a happy number.
- \is_narcissistic(n: integer)\ - Check if n is a narcissistic (Armstrong) number.
- \lcm(a: integer, b: integer)\ - Calculates the Least Common Multiple of two integers.
- \lcm_list(numbers: array)\ - Calculates the LCM of a list of integers.
- \least_common_multiple(a: integer, b: integer)\ - Calculates the least common multiple (LCM) of two integers.
- \log1p(x: number)\ - Calculates the natural logarithm of (1 + x) with high precision.
- \log_base_n(x: number, base: integer)\ - Calculates the logarithm of a number to a specified base N.
- \log_gamma(x: number)\ - Calculates the natural logarithm of the absolute value of Γ(x).
- \look_and_say(seed: string = 1, iterations: integer = 5)\ - Generate the look-and-say sequence.
- \lu_decomposition(matrix: array)\ - Performs LU decomposition of a square matrix (Doolittle algorithm).
- \matrix_add(matrix_a: array, matrix_b: array)\ - Adds two matrices element-wise.
- \matrix_determinant(matrix: array)\ - Computes the determinant of a square matrix.
- \matrix_eigenvalues_2x2(matrix: array)\ - Computes the eigenvalues of a 2×2 matrix using the characteristic equation.
- \matrix_frobenius_norm(matrix: array)\ - Computes the Frobenius norm of a matrix: sqrt(sum of squares of all elements).
- \matrix_infinity_norm(matrix: array)\ - Computes the infinity norm of a matrix: max row-sum of absolute values.
- \matrix_inverse(matrix: array)\ - Computes the inverse of a square matrix.
- \matrix_multiply(matrix_a: array, matrix_b: array)\ - Multiplies two matrices.
- \matrix_one_norm(matrix: array)\ - Computes the 1-norm of a matrix: max column-sum of absolute values.
- \matrix_power(matrix: array, n: integer)\ - Computes the n-th power of a square matrix A^n by repeated multiplication.
- \matrix_rank(matrix: array, tol: number = 1e-10)\ - Computes the rank of a matrix via Gaussian elimination.
- \matrix_scalar_multiply(matrix: array, scalar: integer)\ - Multiplies every element of a matrix by a scalar.
- \matrix_trace(matrix: array)\ - Computes the trace of a square matrix (sum of diagonal elements).
- \matrix_transpose(matrix: array)\ - Transposes a matrix (swaps rows and columns).
- \modulo(number: integer, divisor: integer)\ - Returns the remainder after dividing number by divisor.
- \multinomial_coefficient(args: array = ...)\ - Compute the multinomial coefficient n! / (k1! * k2! * ... * km!).
- \multiplicative_persistence(n: integer)\ - Count iterations to reduce n to a single digit by multiplying its digits.
- \natural_log(x: number)\ - Calculates the natural logarithm (base e) of a number.
- \newton_raphson(f: number, df: number, x0: number, tol: number = 1e-10, max_iter: integer = 100)\ - Finds a root of f(x) = 0 using the Newton-Raphson method.
- \nth_root(x: integer, n: integer)\ - Calculates the nth root of a number.
- \numerical_derivative(f: number, x: number, h: number = 1e-08)\ - Computes the numerical derivative using the central difference method.
- \permutations(n: integer, k: integer)\ - Calculates the number of permutations P(n, k).
- \permutations_with_repetition(n: integer, k: integer)\ - Returns the number of permutations with repetition.
- \polynomial_evaluate(coefficients: array, x: number)\ - Evaluates a polynomial at x using Horner's method.
- \polynomial_roots_cubic(a: number, b: number, c: number, d: number)\ - Finds the three roots of a cubic equation ax³ + bx² + cx + d = 0.
- \polynomial_roots_quadratic(a: number, b: number, c: number)\ - Finds the two roots of a quadratic equation ax² + bx + c = 0.
- \power(base: integer, exponent: integer)\ - Calculates the value of a base raised to a specified exponent.
- \product_list(data: array)\ - Calculates the product of all values in a list.
- \qr_decomposition(matrix: array)\ - Performs QR decomposition via modified Gram-Schmidt orthogonalization.
- \quotient(dividend: integer, divisor: integer)\ - Return the integer portion of a division (truncated toward zero).
- \reduce_to_modulo_range(x: integer, base: integer)\ - Converts a value to a cyclic range using the modulo operation.
- \romberg_integrate(f: number, a: number, b: number, max_order: integer = 10, tol: number = 1e-12)\ - Romberg integration — recursive Richardson extrapolation on the trapezoid rule.
- \runge_kutta_4_step(f: number, t: number, y: number, h: number)\ - Performs one step of the classic 4th-order Runge-Kutta method.
- \safe_division_for_context(numerator: integer, denominator: integer, return_float: boolean = True)\ - Performs division selecting the appropriate operator based on context.
- \safe_sum_with_none(num1: integer, num2: integer)\ - Performs addition of two numbers, safely handling None values.
- \secant_method(f: number, x0: number, x1: number, tol: number = 1e-10, max_iter: integer = 100)\ - Find a root of *f* using the secant method.
- \series_sum(x: number, n: integer, m: integer, coefficients: array)\ - Compute a power series sum: sum(a_i * x^(n + i*m)).
- \sign(x: integer)\ - Returns the sign of a number as -1, 0, or 1.
- \simpsons_integrate(f: number, a: number, b: number, n: integer = 1000)\ - Numerically integrates f(x) from a to b using Simpson's 1/3 rule.
- \solve_linear_system(a: array, b: array)\ - Solves Ax = b via Gaussian elimination with partial pivoting.
- \sqrt_pi(number: number)\ - Calculates the square root of (number * pi).
- \square_pyramidal_number(n: integer)\ - Compute the n-th square pyramidal number = n(n+1)(2n+1)/6.
- \square_root(x: integer)\ - Calculates the square root of a non-negative number.
- \stern_brocot(n: integer)\ - Generate the first n terms of the Stern-Brocot sequence.
- \sum_first_n_cubes(n: integer)\ - Compute 1³ + 2³ + ··· + n³.
- \sum_first_n_powers(n: integer, k: integer)\ - Compute 1^k + 2^k + ··· + n^k.
- \sum_first_n_squares(n: integer)\ - Compute 1² + 2² + ··· + n².
- \sum_x2my2(array_x: array, array_y: array)\ - Returns the sum of the difference of squares of corresponding values.
- \sum_x2py2(array_x: array, array_y: array)\ - Returns the sum of the sum of squares of corresponding values.
- \sum_xmy2(array_x: array, array_y: array)\ - Returns the sum of squares of differences of corresponding values.
- \tetrahedral_number(n: integer)\ - Compute the n-th tetrahedral number T_n = n(n+1)(n+2)/6.
- \trapezoidal_integrate(f: number, a: number, b: number, n: integer = 1000)\ - Numerically integrates f(x) from a to b using the trapezoidal rule.
- \triangular_number(n: integer)\ - Compute the n-th triangular number T_n = n(n+1)/2.
- \true_division(numerator: integer, denominator: integer)\ - Performs "true" (floating-point) division, always returning a float.
- \vector_angle(vec_a: array, vec_b: array)\ - Computes the angle in radians between two vectors.
- \vector_magnitude(vec: array)\ - Computes the Euclidean magnitude (L2 norm) of a vector.
- \vector_normalize(vec: array)\ - Normalizes a vector to unit length (L2 norm = 1).

## fxNumeric.calculator_functions

- \evaluate_expression(expression: string)\ - Safely evaluates a mathematical expression from a string.
- \list_available_constants()\ - Lists all constants available in the expression evaluator.
- \list_available_functions()\ - Lists all functions available in the expression evaluator.

## fxNumeric.complex_functions

- \complex_add(a: number, b: number)\ - Adds two complex numbers represented as (real, imag) tuples.
- \complex_argument(z: number)\ - Computes the argument (phase angle) of a complex number in radians.
- \complex_argument_degrees(z: number)\ - Compute the argument (phase) of a complex number in degrees.
- \complex_conjugate(z: number)\ - Compute the complex conjugate.
- \complex_cos(z: number)\ - Computes cos(z) for complex z.
- \complex_cosh(z: number)\ - Compute the hyperbolic cosine of a complex number.
- \complex_distance(z1: number, z2: number)\ - Compute the distance between two complex numbers.
- \complex_divide(a: number, b: number)\ - Divides complex number a by b.
- \complex_exp(z: number)\ - Computes the complex exponential e^z.
- \complex_from_polar(r: number, theta: number)\ - Create a complex number from polar form.
- \complex_lerp(z1: number, z2: number, t: number)\ - Linear interpolation between two complex numbers.
- \complex_ln(z: number)\ - Computes the principal complex logarithm ln(z).
- \complex_log(z: number)\ - Compute the principal natural logarithm of a complex number.
- \complex_midpoint(z1: number, z2: number)\ - Compute the midpoint between two complex numbers.
- \complex_modulus(z: number)\ - Computes the modulus (absolute value) of a complex number.
- \complex_multiply(a: number, b: number)\ - Multiplies two complex numbers.
- \complex_nth_root(z: number, n: integer, k: integer = 0)\ - Compute the k-th n-th root of a complex number.
- \complex_polar_form(z: number)\ - Convert a complex number to polar form (modulus, argument).
- \complex_power(z: number, w: number)\ - Computes z^w for complex z and w via z^w = e^(w·ln(z)).
- \complex_reciprocal(z: number)\ - Compute the reciprocal (multiplicative inverse) of a complex number.
- \complex_rotate(z: number, angle: number)\ - Rotate a complex number by an angle.
- \complex_sin(z: number)\ - Computes sin(z) for complex z.
- \complex_sinh(z: number)\ - Compute the hyperbolic sine of a complex number.
- \complex_sqrt(z: number)\ - Compute the principal square root of a complex number.
- \complex_subtract(a: number, b: number)\ - Subtracts complex number b from a.
- \complex_tan(z: number)\ - Compute the tangent of a complex number.
- \complex_to_polar(z: number)\ - Converts a complex number from rectangular to polar form.
- \de_moivre(r: number, theta: number, n: integer)\ - Applies De Moivre's theorem: (r e^(iθ))^n = r^n e^(i·n·θ).
- \euler_formula(theta: number)\ - Evaluates Euler's formula e^(iθ) = cos θ + i sin θ.
- \nth_roots_complex(z: number, n: integer)\ - Computes all n-th roots of a complex number z.
- \polar_to_complex(r: number, theta: number)\ - Converts a complex number from polar to rectangular form.
- \roots_of_unity(n: integer)\ - Computes the n-th roots of unity: e^(2πik/n) for k = 0, 1, ..., n-1.

## fxNumeric.conformal_functions

- \cayley_transform(z: number)\ - Cayley transform w = (z - i) / (z + i).
- \exponential_map(z: number)\ - Conformal exponential map w = e^z.
- \inverse_cayley_transform(w: number)\ - Inverse Cayley transform z = i(1 + w) / (1 - w).
- \inverse_joukowski(w: number)\ - Computes both preimages of the Joukowski map: z = (w ± sqrt(w² - 4)) / 2.
- \inversion_map(z: number)\ - Complex inversion w = 1/z.
- \joukowski_transform(z: number)\ - Applies the Joukowski (Zhukovsky) conformal transformation w = z + 1/z.
- \koebe_function(z: number)\ - Koebe function w = z / (1 - z)².
- \log_map(z: number)\ - Principal branch of the complex logarithm w = Log(z).
- \mobius_transform(z: number, a: number, b: number, c: number, d: number)\ - Applies a Möbius (bilinear) transformation w = (az + b) / (cz + d).
- \power_map(z: number, n: number)\ - Conformal power map w = z^n.
- \schwarz_christoffel_half_plane_to_rectangle(z: number, k: number)\ - Approximate Schwarz-Christoffel map from upper half-plane to a rectangle.

## fxNumeric.constants_functions

- \get_constant(name: string)\ - Returns the value of a scientific or mathematical constant by name.
- \list_constants(category: string = None)\ - Lists all available scientific and mathematical constants.

## fxNumeric.conversion_functions

- \acres_to_hectares(acres: number)\ - Convert acres to hectares.
- \add_bool_to_int(boolean_value: boolean, int_value: integer)\ - Adds a boolean value to an integer.
- \adiabatic_index_pressure(p1: integer, v1: integer, v2: integer, gamma: integer)\ - Calculate final pressure in an adiabatic process.
- \adiabatic_temperature(t1: integer, v1: integer, v2: integer, gamma: integer = 1.4)\ - Return the final temperature after adiabatic expansion/compression.
- \angle_convert(value: integer, from_unit: string, to_unit: string)\ - Converts an angle between degrees, radians, and gradians.
- \angular_momentum(mass: integer, velocity: integer, radius: integer)\ - Calculate angular momentum L = m·v·r.
- \antenna_gain_to_effective_area(gain_dbi: number, frequency_hz: number)\ - Convert antenna gain (dBi) to effective aperture area (m²).
- \atmospheres_to_pascals(atm: number)\ - Convert standard atmospheres to pascals.
- \bars_to_pascals(bars: number)\ - Convert bars to pascals.
- \base_to_decimal(text: string, radix: integer)\ - Converts a string representation from the specified base to decimal.
- \bending_moment(force: number, distance: number)\ - Calculates the bending moment M = F × d.
- \bernoulli_velocity(pressure_diff: integer, fluid_density: integer)\ - Calculate flow velocity from pressure difference (Bernoulli's eq.).
- \bin_to_hex(binary: string)\ - Converts a binary string to hexadecimal.
- \bin_to_int(bin_string: string)\ - Converts a binary string to an integer.
- \bin_to_oct(binary: string)\ - Converts a binary string to octal.
- \biot_number(h: integer, l_c: integer, k: integer)\ - Return the Biot number for conductive heat transfer.
- \bmi(weight_kg: integer, height_m: integer)\ - Calculate Body Mass Index (BMI).
- \bool_to_float(value: boolean)\ - Converts a boolean value to a floating-point number.
- \bool_to_int(value: boolean)\ - Converts a boolean value to an integer.
- \boyle_law_volume(p1: integer, v1: integer, p2: integer)\ - Return the new volume using Boyle's law.
- \btu_to_joules(btu: number)\ - Convert British thermal units to joules.
- \bulk_modulus_pressure(bulk_modulus: integer, volume_change: integer, original_volume: integer)\ - Calculate pressure change from bulk modulus.
- \buoyancy_force(fluid_density: integer, displaced_volume: integer, gravity: number = 9.80665)\ - Return the buoyancy force on an immersed object (Archimedes).
- \buoyant_force(fluid_density: integer, displaced_volume: integer, gravity: integer = 9.80665)\ - Calculate buoyant force using Archimedes' principle: F = ρ * V * g.
- \bytes_to_human_readable(size_bytes: integer)\ - Convert bytes to human-readable string (B, KB, MB, GB, TB).
- \bytes_to_int(byte_sequence: any, byteorder: string = big)\ - Converts a byte sequence to an integer.
- \calories_to_joules(cal: number)\ - Convert thermochemical calories to joules.
- \candela_to_lumens(cd: number, solid_angle_sr: number = 12.566370614359172)\ - Convert luminous intensity (candela) to luminous flux (lumens).
- \capacitance_parallel_plate(area: integer, distance: integer, epsilon_r: integer = 1.0)\ - Calculate parallel plate capacitance C = ε₀·εᵣ·A/d.
- \capacitive_reactance(frequency: integer, capacitance: integer)\ - Return the capacitive reactance of a capacitor.
- \capacitor_energy(capacitance: integer, voltage: integer)\ - Calculate energy stored in a capacitor: E = 0.5 * C * V².
- \centripetal_acceleration(velocity: integer, radius: integer)\ - Calculate centripetal acceleration a = v² / r.
- \centripetal_force(mass: integer, velocity: integer, radius: integer)\ - Calculate centripetal force F = m·v²/r.
- \charles_law_volume(v1: integer, t1: integer, t2: integer)\ - Return the new volume using Charles's law.
- \cmyk_to_rgb(c: number, m: number, y: number, k: number)\ - Convert CMYK colour to RGB.
- \color_contrast_ratio(r1: integer, g1: integer, b1: integer, r2: integer, g2: integer, b2: integer)\ - WCAG 2.1 contrast ratio between two sRGB colours.
- \color_luminance(r: integer, g: integer, b: integer)\ - Relative luminance of an sRGB colour per WCAG 2.1.
- \color_temperature_to_rgb(kelvin: number)\ - Convert a colour temperature (Kelvin) to an approximate RGB tuple.
- \compton_wavelength_shift(angle: integer)\ - Return the Compton wavelength shift for photon scattering.
- \convert_bytes_to(unit: string, value: integer)\ - Converts a byte value to a specified target unit.
- \convert_string_to_float_with_locale(number_string: string, target_locale: string = None)\ - Converts a numeric string to a float, interpreting decimal and thousands separators
- \convert_units(number: integer, from_unit: string, to_unit: string)\ - Convert a value from one measurement unit to another.
- \coordinate_decimal_to_dms(decimal_degrees: integer, is_longitude: boolean = False)\ - Converts decimal degrees to DMS (degrees, minutes, seconds, direction).
- \coordinate_dms_to_decimal(degrees: integer, minutes: integer, seconds: integer, direction: string = N)\ - Converts geographic coordinates from DMS to decimal degrees.
- \coulomb_force(charge1: integer, charge2: integer, distance: integer)\ - Calculate electrostatic force: F = k * |q₁·q₂| / r².
- \coulombs_force(q1: integer, q2: integer, distance: integer)\ - Return the electrostatic force between two point charges.
- \coulombs_law(q1: integer, q2: integer, distance: integer)\ - Computes electrostatic force via Coulomb's law.
- \cubic_meters_per_second_to_liters_per_minute(cms: number)\ - Convert cubic meters per second to liters per minute.
- \current_divider(total_current: integer, r_branch: integer, r_other: integer)\ - Return the branch current in a two-resistor current divider.
- \db_to_power(decibels: integer)\ - Convert decibels to a power ratio.
- \db_to_voltage(decibels: integer)\ - Convert decibels to a voltage/amplitude ratio.
- \de_broglie_wavelength(mass: integer, velocity: integer)\ - Calculate de Broglie wavelength: λ = h / (m * v).
- \dec_to_oct(number: integer)\ - Converts a decimal integer to octal.
- \decibel_add(db1: integer, db2: integer)\ - Add two sound levels in decibels.
- \decibel_sum(db_a: number, db_b: number)\ - Logarithmic addition of two sound levels in decibels.
- \decimal_to_fraction(number: number, max_denominator: integer = 1000)\ - Convert a decimal number to a fraction (numerator, denominator).
- \degrees_to_gradians(deg: number)\ - Convert degrees to gradians (gon).
- \density_to_specific_gravity(density: number, reference: number = 997.0)\ - Convert density to specific gravity relative to a reference.
- \dew_point(temperature_c: integer, relative_humidity: integer)\ - Calculate dew point temperature using the Magnus formula.
- \dft_magnitude(signal: array, k: integer)\ - Computes the magnitude of the k-th DFT coefficient (standalone, no FFT).
- \dft_phase(signal: array, k: integer)\ - Computes the phase (in radians) of the k-th DFT coefficient.
- \doppler_freq(source_freq: integer, velocity: integer, wave_speed: integer = 343.0)\ - Calculate observed frequency due to the Doppler effect.
- \doppler_frequency(source_freq: integer, wave_speed: integer, observer_speed: integer = 0.0, source_speed: integer = 0.0)\ - Return the observed frequency due to the Doppler effect.
- \doppler_shift(source_frequency: integer, relative_velocity: integer, wave_speed: integer = 343.0)\ - Calculate observed frequency due to Doppler effect (source moving).
- \drag_force(drag_coefficient: integer, fluid_density: integer, velocity: integer, reference_area: integer)\ - Calculate aerodynamic drag force: F_d = ½·C_d·ρ·v²·A.
- \drift_velocity(current: integer, n_density: integer, charge: integer, area: integer)\ - Calculate charge carrier drift velocity v = I / (n·q·A).
- \elastic_modulus(stress: integer, strain: integer)\ - Calculate Young's modulus (elastic modulus): E = stress / strain.
- \elastic_potential_energy(k: integer, x: integer)\ - Calculate elastic potential energy U = ½·k·x².
- \electric_field_parallel_plate(voltage: integer, distance: integer)\ - Calculate uniform electric field between parallel plates E = V/d.
- \electric_field_point(charge: integer, distance: integer)\ - Calculate the electric field magnitude from a point charge: E = k·|q|/r².
- \electric_potential(charge: integer, distance: integer)\ - Calculate electric potential V = k·q/r.
- \energy_convert(value: integer, from_unit: string, to_unit: string)\ - Converts an energy value between common units.
- \escape_velocity(mass: integer, radius: integer)\ - Computes the escape velocity from a celestial body.
- \float_to_int_truncated(number: number)\ - Converts a float to an integer by truncating the decimal part (towards zero).
- \float_to_json_safe(float_value: number)\ - Converts a floating-point number to a JSON-safe representation.
- \focal_length(object_distance: integer, image_distance: integer)\ - Calculate focal length using the thin lens equation: 1/f = 1/do + 1/di.
- \foot_pounds_to_newton_meters(ftlb: number)\ - Convert foot-pounds to newton-metres (torque).
- \fraction_to_decimal(numerator: integer, denominator: integer)\ - Convert a fraction to its decimal representation.
- \free_space_path_loss(freq: integer, distance: integer)\ - Calculate free-space path loss in decibels.
- \freq_from_wavelength(wavelength_m: integer, velocity: integer = 299792458.0)\ - Calculate frequency from wavelength.
- \frequency_to_wavelength(frequency_hz: number)\ - Convert electromagnetic frequency to wavelength.
- \friction_force(normal_force: integer, coefficient: integer)\ - Calculate friction force: F_f = μ * N.
- \friis_transmission(pt: integer, gt: integer, gr: integer, wavelength_m: integer, distance: integer)\ - Calculate received power using the Friis transmission equation.
- \froude_number(v: integer, g: integer, length: integer)\ - Calculate the Froude number Fr = v / √(g · L).
- \gallons_to_liters(gallons: number, us: boolean = True)\ - Convert gallons to litres.
- \gay_lussac_pressure(p1: integer, t1: integer, t2: integer)\ - Return the new pressure using Gay-Lussac's law.
- \gradians_to_degrees(grad: number)\ - Convert gradians (gon) to degrees.
- \grams_to_troy_ounces(grams: number)\ - Convert grams to troy ounces.
- \grashof_number(gravity: integer, beta_coeff: integer, delta_temp: integer, length: integer, kinematic_viscosity: integer)\ - Return the Grashof number for natural convection.
- \gravitational_force(m1: integer, m2: integer, distance: integer)\ - Computes gravitational force via Newton's law.
- \gravitational_potential_energy(mass: integer, height: integer, gravity: integer = 9.80665)\ - Calculate gravitational potential energy: U = m·g·h.
- \h_concentration_to_ph(concentration: number)\ - Convert hydrogen ion concentration [H⁺] to pH.
- \half_life_remaining(initial_quantity: integer, half_life: integer, elapsed_time: integer)\ - Calculate remaining quantity after radioactive/exponential decay.
- \heat_capacity(mass: integer, specific_heat: integer, temperature_change: integer)\ - Calculate heat energy Q = m·c·ΔT.
- \heat_engine_efficiency(t_hot: integer, t_cold: integer)\ - Return the Carnot efficiency of a heat engine.
- \heat_index(temperature_c: integer, relative_humidity: integer)\ - Calculate heat index (feels-like temperature) in Celsius.
- \heat_transfer_conduction(thermal_conductivity: integer, area: integer, temperature_diff: integer, thickness: integer)\ - Calculate heat transfer rate by conduction (Fourier's law): Q = k·A·ΔT/L.
- \hectares_to_acres(hectares: number)\ - Convert hectares to acres.
- \hertz_to_rpm(hz: number)\ - Convert hertz to revolutions per minute.
- \hex_to_bin(hexadecimal: string)\ - Converts a hexadecimal string to binary.
- \hex_to_int(hex_string: string)\ - Converts a hexadecimal string to an integer.
- \hex_to_oct(hexadecimal: string)\ - Converts a hexadecimal string to octal.
- \hookes_law(spring_constant: integer, displacement: integer)\ - Calculate the restoring force of a spring using Hooke's law: F = -k * x.
- \horsepower_to_watts(hp: number)\ - Convert mechanical horsepower to watts.
- \hsl_to_rgb(h: number, s: number, lightness: number)\ - Convert HSL colour to RGB.
- \hsv_to_rgb(h: number, s: number, v: number)\ - Convert HSV colour to RGB.
- \ideal_gas_pressure(moles: integer, temperature_k: integer, volume_m3: integer)\ - Calculate pressure using the ideal gas law: P = nRT / V.
- \ieee754_hex_representation(numero: number)\ - Returns the hexadecimal representation of a floating-point number
- \impedance_parallel(z1: any, z2: any)\ - Calculate parallel combination of two impedances.
- \impedance_series(r: integer, l: integer, c: integer, freq: integer)\ - Calculate impedance of a series RLC circuit.
- \impedance_series_rlc(resistance: integer, inductive_react: integer, capacitive_react: integer)\ - Return the total impedance of a series RLC circuit.
- \impulse(force: integer, time_seconds: integer)\ - Calculate impulse: J = F * Δt.
- \inductive_reactance(frequency: integer, inductance: integer)\ - Return the inductive reactance of an inductor.
- \inductor_energy(inductance: integer, current: integer)\ - Calculate energy stored in an inductor: E = ½·L·I².
- \int_to_binary_clean(number: integer)\ - Converts an integer to its binary representation as a string, without the '0b' prefix.
- \int_to_binary_with_prefix(number: integer)\ - Converts an integer to its binary representation as a string, including the '0b' prefix.
- \int_to_bytes(integer_value: integer, num_bytes: integer, byteorder: string = big)\ - Converts an integer to a byte sequence.
- \int_to_float(number: integer)\ - Converts an integer to a floating-point number.
- \int_to_hex_clean(number: integer)\ - Converts an integer to its hexadecimal representation as a string, without the '0x' prefix.
- \int_to_hex_with_prefix(number: integer)\ - Converts an integer to its hexadecimal representation as a string, including the '0x' prefix.
- \int_to_octal_with_prefix(number: integer)\ - Converts an integer to its octal representation as a string, including the '0o' prefix.
- \int_to_roman(number: integer)\ - Convert a positive integer to a Roman numeral string.
- \is_numeric_type(value: any)\ - Checks if a value is of numeric type.
- \joules_to_btu(joules: number)\ - Convert joules to British thermal units.
- \joules_to_calories(joules: number)\ - Convert joules to thermochemical calories.
- \joules_to_kwh(joules: number)\ - Convert joules to kilowatt-hours.
- \kilometers_to_light_years(km: number)\ - Convert kilometres to light-years.
- \kinematic_displacement(initial_velocity: integer, time: integer, acceleration: integer)\ - Calculate displacement using kinematic equation s = v₀·t + ½·a·t².
- \kinetic_energy(mass: integer, velocity: integer)\ - Computes kinetic energy.
- \kinetic_energy_relativistic(rest_mass: integer, velocity: integer)\ - Calculate relativistic kinetic energy: KE = (γ - 1)·m·c².
- \kmh_to_knots(kmh: number)\ - Convert kilometres per hour to knots.
- \knots_to_kmh(knots: number)\ - Convert knots to kilometres per hour.
- \knudsen_number(mean_free_path: number, length: number)\ - Compute the Knudsen number for rarefied gas dynamics.
- \kwh_to_joules(kwh: number)\ - Convert kilowatt-hours to joules.
- \lens_magnification(image_distance: integer, object_distance: integer)\ - Calculate the magnification of a thin lens: M = -di / do.
- \light_years_to_kilometers(ly: number)\ - Convert light-years to kilometres.
- \liters_per_minute_to_cubic_meters_per_second(lpm: number)\ - Convert liters per minute to cubic meters per second.
- \liters_to_gallons(liters: number, us: boolean = True)\ - Convert litres to gallons.
- \lorentz_factor(velocity: integer)\ - Calculate the Lorentz factor: γ = 1 / √(1 - v²/c²).
- \lumens_to_candela(lm: number, solid_angle_sr: number = 12.566370614359172)\ - Convert luminous flux (lumens) to luminous intensity (candela).
- \luminous_flux(candela: number, solid_angle_sr: number = 12.566370614359172)\ - Convert luminous intensity (candela) to luminous flux (lumens).
- \mach_number(velocity: integer, speed_of_sound: integer = 343.0)\ - Calculate the Mach number: M = v / c.
- \mach_to_ms(mach: number, speed_of_sound: number = 343.0)\ - Convert Mach number to metres per second.
- \magnetic_field_solenoid(n_turns: integer, current: integer, length: integer)\ - Calculate magnetic field inside a solenoid B = μ₀·(N/L)·I.
- \magnetic_flux(magnetic_field: integer, area: integer, angle: integer = 0.0)\ - Calculate magnetic flux: Φ = B·A·cos(θ).
- \magnetic_flux_density(permeability: integer, current: integer, distance: integer)\ - Return the magnetic flux density around a long straight wire.
- \magnetic_force_charge(charge: integer, velocity: integer, magnetic_field: integer, angle: integer = 1.5707963267948966)\ - Calculate magnetic force on a moving charge F = q·v·B·sin(θ).
- \magnetic_force_on_wire(current: integer, length: integer, magnetic_field: integer, angle: integer = 1.5707963267948966)\ - Calculate force on a current-carrying wire: F = I·L·B·sin(θ).
- \moment_of_inertia_point(mass: integer, radius: integer)\ - Calculate moment of inertia for a point mass I = m·r².
- \momentum(mass: integer, velocity: integer)\ - Calculate linear momentum: p = m * v.
- \ms_to_mach(ms: number, speed_of_sound: number = 343.0)\ - Convert metres per second to Mach number.
- \nernst_potential(z: integer, concentration_out: integer, concentration_in: integer, temperature: integer = 310.15)\ - Return the Nernst equilibrium potential for an ion.
- \newton_meters_to_foot_pounds(nm: number)\ - Convert newton-metres to foot-pounds (torque).
- \newtons_to_pounds_force(newtons: number)\ - Convert newtons to pounds-force.
- \noise_figure_to_temperature(nf_db: number, t0: number = 290.0)\ - Convert noise figure (dB) to equivalent noise temperature (Kelvin).
- \number_to_base(number: integer, radix: integer, min_length: integer = 0)\ - Converts a decimal integer to a string in the specified base.
- \number_to_bool(number: integer)\ - Converts an integer or float to a boolean value.
- \number_to_complex(number: integer)\ - Converts an integer or float to a complex number with zero imaginary part.
- \number_to_hexadecimal(number_input: integer)\ - Returns a string representing the hexadecimal value of a number.
- \number_to_octal(number_input: integer)\ - Returns a string representing the octal value of a number.
- \number_to_string(number: integer)\ - Converts an integer or float to its string representation.
- \number_to_words(number: integer, lang: string = en)\ - Convert an integer to its written-word representation.
- \nusselt_number(h: number, length: number, k: number)\ - Compute the Nusselt number for convective heat transfer.
- \nyquist_frequency(sample_rate: number)\ - Returns the Nyquist frequency for a given sample rate.
- \oct_to_bin(octal: string)\ - Converts an octal string to binary.
- \oct_to_hex(octal: string)\ - Converts an octal string to hexadecimal.
- \octal_to_int(octal_string: string)\ - Converts an octal string to an integer.
- \ohms_law(voltage: integer = None, current: integer = None, resistance: integer = None)\ - Solves Ohm's law V = I × R for the missing variable.
- \orbital_period(semi_major_axis: integer, central_mass: integer)\ - Calculate orbital period using Kepler's third law.
- \parallel_resistance(r1: integer, r2: integer)\ - Return the equivalent resistance of two resistors in parallel.
- \pascals_to_atmospheres(pa: number)\ - Convert pascals to standard atmospheres.
- \pascals_to_bars(pa: number)\ - Convert pascals to bars.
- \pascals_to_psi(pa: number)\ - Convert pascals to pounds per square inch (PSI).
- \pendulum_period(length: integer, gravity: integer = 9.80665)\ - Calculate the period of a simple pendulum: T = 2π√(L/g).
- \ph_to_h_concentration(ph: number)\ - Convert pH to hydrogen ion concentration [H⁺].
- \photon_energy(frequency: integer)\ - Calculate the energy of a photon: E = h * f.
- \photon_momentum(wavelength: integer)\ - Calculate photon momentum p = h/λ.
- \planck_radiation_peak(temperature: integer)\ - Return the spectral radiance at the Wien peak wavelength.
- \potential_energy(mass: integer, height: integer, g: integer = 9.80665)\ - Computes gravitational potential energy.
- \pounds_force_to_newtons(lbf: number)\ - Convert pounds-force to newtons.
- \power_mechanical(work: integer, time: integer)\ - Calculate mechanical power P = W / t.
- \power_physics(work: integer, time_seconds: integer)\ - Calculate mechanical power: P = W / t.
- \power_to_db(ratio: integer)\ - Convert a power ratio to decibels.
- \prandtl_number(cp: integer, mu: integer, k: integer)\ - Calculate the Prandtl number Pr = Cp · μ / k.
- \pressure_at_depth(fluid_density: integer, depth: integer, gravity: integer = 9.80665, surface_pressure: integer = 101325.0)\ - Calculate pressure at a depth in a fluid: P = P₀ + ρ·g·h.
- \pressure_convert(value: integer, from_unit: string, to_unit: string)\ - Converts a pressure value between common units.
- \projectile_max_height(velocity: integer, angle: integer, gravity: integer = 9.80665)\ - Calculate the maximum height of a projectile: H = v²·sin²(θ) / (2g).
- \projectile_range(velocity: integer, angle: integer, gravity: integer = 9.80665)\ - Calculate the horizontal range of a projectile (flat ground).
- \projectile_time_of_flight(velocity: integer, angle: integer, gravity: integer = 9.80665)\ - Calculate time of flight for a projectile: T = 2v·sin(θ) / g.
- \psi_to_pascals(psi: number)\ - Convert pounds per square inch (PSI) to pascals.
- \radiation_dose_convert(value: number, from_unit: string, to_unit: string)\ - Convert between radiation dose units.
- \rc_time_constant(resistance: integer, capacitance: integer)\ - Calculate the RC time constant: τ = R·C.
- \relativistic_energy(mass: integer, velocity: integer)\ - Calculate total relativistic energy E = γ·m·c².
- \resistivity_resistance(resistivity: integer, length: integer, area: integer)\ - Calculate resistance from resistivity R = ρ·L/A.
- \resistors_parallel_pair(r1: integer, r2: integer)\ - Calculate equivalent resistance of two resistors in parallel.
- \resonant_freq(l: integer, c: integer)\ - Calculate resonant frequency of an LC circuit.
- \reynolds_number(density: integer, velocity: integer, length: integer, viscosity: integer)\ - Calculate the Reynolds number: Re = ρ * v * L / μ.
- \rgb_to_cmyk(r: integer, g: integer, b: integer)\ - Convert RGB colour to CMYK (Cyan, Magenta, Yellow, Key/Black).
- \rgb_to_hsl(r: integer, g: integer, b: integer)\ - Convert RGB colour to HSL.
- \rgb_to_hsv(r: integer, g: integer, b: integer)\ - Convert RGB colour to HSV (Hue, Saturation, Value).
- \rl_time_constant(inductance: integer, resistance: integer)\ - Return the RL circuit time constant.
- \rms_voltage(peak_voltage: integer)\ - Return the root-mean-square voltage from a sinusoidal peak.
- \roman_to_int(roman: string)\ - Convert a Roman numeral string to an integer.
- \round_float_to_int(number: number)\ - Rounds a float to the nearest integer.
- \rpm_to_hertz(rpm: number)\ - Convert revolutions per minute to hertz.
- \safe_convert_number(value: any, default_value: integer = 0, target_type: any = <class 'float'>)\ - Safely converts an input value to a numeric type (float or int), using a
- \schwarzschild_radius(mass: integer)\ - Calculate the Schwarzschild radius: r_s = 2GM / c².
- \shannon_capacity(bandwidth: integer, snr: integer)\ - Calculate Shannon channel capacity.
- \signal_rms(samples: array)\ - Computes the RMS (Root Mean Square) of a discrete signal.
- \signal_snr_db(signal_power: number, noise_power: number)\ - Computes the signal-to-noise ratio in decibels.
- \signal_to_noise_ratio(signal_power: number, noise_power: number)\ - Compute the signal-to-noise ratio in decibels.
- \skin_depth(freq: integer, resistivity: integer, permeability: integer = 1.2566370614359173e-06)\ - Calculate skin depth in a conductor.
- \snells_law(n1: integer, theta1: integer, n2: integer)\ - Computes the refraction angle via Snell's law.
- \snells_law_angle(n1: integer, n2: integer, theta1: integer)\ - Calculate refracted angle using Snell's law: n₁·sin(θ₁) = n₂·sin(θ₂).
- \sound_intensity_level(intensity: integer, reference: integer = 1e-12)\ - Calculate sound intensity level in decibels.
- \specific_heat_energy(mass: integer, specific_heat: integer, temperature_change: integer)\ - Calculate thermal energy: Q = m * c * ΔT.
- \specific_impulse(thrust: integer, mass_flow_rate: integer, gravity: number = 9.80665)\ - Return the specific impulse of a rocket engine.
- \speed_convert(value: integer, from_unit: string, to_unit: string)\ - Converts a speed value between common units.
- \spring_potential_energy(spring_constant: integer, displacement: integer)\ - Calculate elastic potential energy: U = ½·k·x².
- \stefan_boltzmann_power(emissivity: integer, area: integer, temperature: integer)\ - Calculate radiated power using Stefan-Boltzmann law: P = ε·σ·A·T⁴.
- \stefan_boltzmann_temperature(power: number, emissivity: number, area: number)\ - Calculates temperature from radiated power (inverse Stefan-Boltzmann).
- \strain(change_in_length: number, original_length: number)\ - Calculates engineering strain ε = ΔL / L₀.
- \stress(force: number, area: number)\ - Calculates mechanical stress σ = F / A.
- \strouhal_number(freq: number, length: number, velocity: number)\ - Compute the Strouhal number for oscillating flow.
- \temperature_convert(value: integer, from_unit: string, to_unit: string)\ - Converts a temperature between Celsius, Fahrenheit, and Kelvin.
- \terminal_velocity(mass: integer, drag_coefficient: integer, area: integer, air_density: integer = 1.225)\ - Calculate terminal velocity of a falling object.
- \thermal_expansion_length(original_length: integer, alpha: integer, delta_temp: integer)\ - Return the change in length due to thermal expansion.
- \thermal_noise(bandwidth: integer, temperature: integer = 290.0)\ - Calculate Johnson-Nyquist thermal noise power.
- \thermal_resistance(length: integer, thermal_conductivity: integer, area: integer)\ - Calculate thermal resistance for conduction: R_th = L / (k·A).
- \to_js_safe_integer(number: integer)\ - Converts a number to a JavaScript-safe integer.
- \torque(force: integer, lever_arm: integer, angle: integer = 1.5707963267948966)\ - Calculate torque τ = F·r·sin(θ).
- \troy_ounces_to_grams(oz_t: number)\ - Convert troy ounces to grams.
- \voltage_divider(vin: integer, r1: integer, r2: integer)\ - Return the output voltage of a resistive voltage divider.
- \voltage_to_db(ratio: integer)\ - Convert a voltage/amplitude ratio to decibels.
- \watts_to_horsepower(watts: number)\ - Convert watts to mechanical horsepower.
- \wave_frequency(wave_speed: integer, wavelength: integer)\ - Calculate wave frequency f = v/λ.
- \wave_speed(frequency: integer, wavelength_m: integer)\ - Calculate wave speed: v = f * λ.
- \wavelength(freq: integer, velocity: integer = 299792458.0)\ - Calculate wavelength from frequency.
- \wavelength_to_frequency(wavelength_m: number)\ - Convert electromagnetic wavelength to frequency.
- \weber_number(rho: integer, v: integer, length: integer, sigma: integer)\ - Calculate the Weber number We = ρ · v² · L / σ.
- \wien_displacement_peak(temperature: integer)\ - Return the peak wavelength from Wien's displacement law.
- \wind_chill(temperature_c: integer, wind_speed_kmh: integer)\ - Calculate wind chill index using the North American formula.
- \work_done(force: integer, displacement: integer, angle: integer = 0.0)\ - Calculate work done: W = F·d·cos(θ).

## fxNumeric.coordinate_systems_functions

- \bipolar_to_cartesian(u: number, v: number, a: number = 1.0)\ - Converts bipolar (u, v) to Cartesian (x, y).
- \cartesian_to_cylindrical(x: number, y: number, z: number)\ - Converts Cartesian (x, y, z) to cylindrical (rho, phi, z).
- \cartesian_to_parabolic_cylindrical(x: number, y: number, z: number)\ - Converts Cartesian (x, y, z) to parabolic cylindrical (u, v, z).
- \cylindrical_to_cartesian(rho: number, phi: number, z: number)\ - Converts cylindrical (rho, phi, z) to Cartesian (x, y, z).
- \elliptic_cylindrical_to_cartesian(u: number, v: number, z: number, a: number = 1.0)\ - Converts elliptic cylindrical (u, v, z) to Cartesian (x, y, z).
- \oblate_spheroidal_to_cartesian(xi: number, eta: number, phi: number, a: number = 1.0)\ - Converts oblate spheroidal (ξ, η, φ) to Cartesian (x, y, z).
- \parabolic_cylindrical_to_cartesian(u: number, v: number, z: number)\ - Converts parabolic cylindrical (u, v, z) to Cartesian (x, y, z).
- \paraboloidal_to_cartesian(u: number, v: number, phi: number)\ - Converts paraboloidal (u, v, phi) to Cartesian (x, y, z).
- \prolate_spheroidal_to_cartesian(xi: number, eta: number, phi: number, a: number = 1.0)\ - Converts prolate spheroidal (ξ, η, φ) to Cartesian (x, y, z).
- \toroidal_to_cartesian(u: number, v: number, phi: number, a: number = 1.0)\ - Converts toroidal (u, v, φ) to Cartesian (x, y, z).

## fxNumeric.curves_functions

- \arc_length_function(f: number, a: number, b: number, n: integer = 10000)\ - Computes the arc length of y = f(x) over [a, b].
- \arc_length_parametric(x_func: number, y_func: number, t_start: number, t_end: number, n: integer = 10000)\ - Computes the arc length of a parametric curve (x(t), y(t)) over [t_start, t_end].
- \arc_length_polar(r_func: number, theta_start: number, theta_end: number, n: integer = 10000)\ - Computes the arc length of a polar curve r = f(θ) over [θ_start, θ_end].
- \archimedean_spiral(t: number, a: number = 0.0, b: number = 1.0)\ - Returns the (x, y) coordinates of an Archimedean spiral at parameter t.
- \astroid(t: number, a: number = 1.0)\ - Returns the (x, y) coordinates of an astroid at parameter t.
- \bezier_cubic(p0: any, p1: any, p2: any, p3: any, t: number)\ - Evaluate a cubic Bézier curve at parameter t.
- \bezier_quadratic(p0: any, p1: any, p2: any, t: number)\ - Evaluate a quadratic Bézier curve at parameter t.
- \brachistochrone_time(x_end: number, y_end: number, g: number = 9.80665)\ - Approximate the brachistochrone (fastest descent) time.
- \butterfly_curve(t: number)\ - Evaluate a point on the butterfly curve.
- \cardioid(t: number, a: number = 1.0)\ - Returns the (x, y) coordinates of a cardioid at parameter t.
- \cassini_oval(t: number, a: number = 1.0, c: number = 1.2)\ - Cassini oval in polar form: r² = a² cos(2t) ± √(a⁴cos²(2t) - a⁴ + c⁴).
- \catenary(x: number, a: number = 1.0)\ - Evaluates the catenary curve y = a * cosh(x / a) at point x.
- \catmull_rom(p0: number, p1: number, p2: number, p3: number, t: number)\ - Catmull-Rom spline interpolation between p1 and p2.
- \cissoid(t: number, a: number = 1.0)\ - Returns the (x, y) coordinates of the cissoid of Diocles at parameter t.
- \cochleoid(a: number, theta: number)\ - Compute r for the cochleoid curve: r = a·sin(θ)/θ.
- \conchoid_of_nicomedes(t: number, a: number = 1.0, b: number = 2.0)\ - Conchoid of Nicomedes: r = a/cos(t) + b, returned as (x, y).
- \cornu_spiral(t: number, n: integer = 50)\ - Evaluate the Cornu (Euler) spiral at parameter t.
- \curvature_explicit(f: number, x: number, h: number = 1e-05)\ - Computes the curvature κ of y = f(x) at point x using numerical derivatives.
- \curvature_parametric(x_func: number, y_func: number, t: number, h: number = 1e-05)\ - Computes the curvature κ of a parametric curve (x(t), y(t)) at parameter t.
- \curvature_polar(r_func: number, theta: number, h: number = 1e-05)\ - Computes the curvature κ of a polar curve r = f(θ) at angle θ.
- \cycloid(t: number, r: number = 1.0)\ - Returns the (x, y) coordinates of a cycloid at parameter t.
- \cycloid_arc_length(r: number)\ - Computes the arc length of one arch of a cycloid with radius r.
- \deltoid(t: number, a: number = 1.0)\ - Deltoid (3-cusped hypocycloid): x = a(2cos(t) + cos(2t)), y = a(2sin(t) - sin(2t)).
- \epicycloid(t: number, R: number, r: number)\ - Returns the (x, y) coordinates of an epicycloid at parameter t.
- \epitrochoid(R: number, r: number, d: number, t: number)\ - Evaluate an epitrochoid curve point.
- \evolute_ellipse(a: number, b: number, t: number)\ - Compute a point on the evolute of an ellipse.
- \fermat_spiral(a: number, theta: number)\ - Compute r for Fermat's spiral: r = a·√θ.
- \folium_of_descartes(t: number, a: number = 1.0)\ - Returns the (x, y) coordinates of the folium of Descartes at parameter t.
- \gaussian_curvature_cylinder(r: number)\ - Returns the Gaussian curvature of a circular cylinder of radius r.
- \gaussian_curvature_sphere(r: number)\ - Returns the Gaussian curvature of a sphere of radius r.
- \gaussian_curvature_torus(R: number, r: number, theta: number)\ - Returns the Gaussian curvature of a torus at angle theta.
- \hermite_interpolation(p0: number, p1: number, m0: number, m1: number, t: number)\ - Cubic Hermite interpolation between two points.
- \hypocycloid(t: number, R: number, r: number)\ - Returns the (x, y) coordinates of a hypocycloid at parameter t.
- \hypotrochoid(R: number, r: number, d: number, t: number)\ - Evaluate a hypotrochoid curve point.
- \involute_of_circle(t: number, r: number = 1.0)\ - Returns the (x, y) coordinates of the involute of a circle at parameter t.
- \lemniscate(t: number, a: number = 1.0)\ - Returns the (x, y) coordinates of the lemniscate of Bernoulli at parameter t.
- \limacon(t: number, a: number = 1.0, b: number = 0.5)\ - Limaçon of Pascal: r = a + b·cos(t), returned as (x, y).
- \lissajous(t: number, a: number = 1.0, b: number = 1.0, omega_x: number = 3.0, omega_y: number = 2.0, delta: number = 0.0)\ - Lissajous figure: x = a·sin(ωₓt + δ), y = b·sin(ωᵧt).
- \lituus(a: number, theta: number)\ - Compute r for the lituus curve: r = a / √θ.
- \logarithmic_spiral(t: number, a: number = 1.0, b: number = 0.2)\ - Returns the (x, y) coordinates of a logarithmic spiral at parameter t.
- \mean_curvature_sphere(r: number)\ - Returns the mean curvature of a sphere of radius r.
- \nephroid(t: number, a: number = 1.0)\ - Nephroid curve: x = a(3cos(t) - cos(3t)), y = a(3sin(t) - sin(3t)).
- \piriform(t: number, a: number = 1.0, b: number = 1.0)\ - Piriform (pear-shaped) curve: x = a(1 + sin(t)), y = b·cos(t)(1 + sin(t)).
- \principal_curvatures_ellipsoid(a: number, b: number, c: number, u: number, v: number)\ - Computes the principal curvatures of an ellipsoid at surface point (u, v).
- \radius_of_curvature(f: number, x: number, h: number = 1e-05)\ - Computes the radius of curvature R = 1/κ of y = f(x) at point x.
- \rose_curve(t: number, a: number = 1.0, k: integer = 3)\ - Returns the (x, y) coordinates of a rose (rhodonea) curve at parameter t.
- \strophoid(t: number, a: number = 1.0)\ - Returns the (x, y) coordinates of a right strophoid at parameter t.
- \superellipse(a: number, b: number, n_exp: number, t: number)\ - Evaluate a point on a superellipse.
- \surface_area_revolution(f: number, a: number, b: number, n: integer = 1000)\ - Computes the surface area of revolution of y = f(x) around the x-axis.
- \tautochrone_period(radius: number, g: number = 9.80665)\ - Compute the period of a tautochrone (cycloid) pendulum.
- \tractrix(t: number, a: number = 1.0)\ - Returns the (x, y) coordinates of a tractrix at parameter t.
- \witch_of_agnesi(x: number, a: number = 1.0)\ - Evaluates the Witch of Agnesi curve y = 8a^3 / (x^2 + 4a^2) at point x.

## fxNumeric.distribution_functions

- \beta_dist(x: any, alpha: any, beta: any, cumulative: any = True)\ - Beta distribution PDF or CDF.
- \beta_inv(probability: any, alpha: any, beta: any)\ - Inverse of the Beta cumulative distribution.
- \binom_dist(number_s: any, trials: any, probability_s: any, cumulative: any = True)\ - Binomial distribution PMF or CDF.
- \binom_dist_range(trials: any, probability_s: any, number_s: any, number_s2: any = None)\ - Probability of a binomial trial result between two values.
- \binom_inv(trials: any, probability_s: any, alpha: any)\ - Inverse binomial distribution (CRITBINOM).
- \chisq_dist(x: any, deg_freedom: any, cumulative: any = True)\ - Chi-squared distribution PDF or CDF.
- \chisq_dist_rt(x: any, deg_freedom: any)\ - Right-tailed Chi-squared distribution.
- \chisq_inv(probability: any, deg_freedom: any)\ - Inverse of the chi-squared cumulative distribution.
- \chisq_inv_rt(probability: any, deg_freedom: any)\ - Inverse of the right-tailed Chi-squared distribution.
- \expon_dist(x: any, lambda_: any, cumulative: any = True)\ - Exponential distribution PDF or CDF.
- \f_dist(x: any, deg_freedom1: any, deg_freedom2: any, cumulative: any = True)\ - F-distribution PDF or CDF.
- \f_dist_rt(x: any, deg_freedom1: any, deg_freedom2: any)\ - Right-tailed F-distribution.
- \f_inv(probability: any, deg_freedom1: any, deg_freedom2: any)\ - Inverse of the F cumulative distribution.
- \f_inv_rt(probability: any, deg_freedom1: any, deg_freedom2: any)\ - Inverse of the right-tailed F-distribution.
- \gamma_dist(x: any, alpha: any, beta: any, cumulative: any = True)\ - Gamma distribution PDF or CDF.
- \gamma_inv(probability: any, alpha: any, beta: any)\ - Inverse of the Gamma cumulative distribution.
- \gauss(z: any)\ - Probability that a standard normal variable falls between 0 and z.
- \hypgeom_dist(sample_s: any, number_sample: any, population_s: any, number_pop: any, cumulative: any = True)\ - Hypergeometric distribution PMF or CDF.
- \lognorm_dist(x: any, mean: any, standard_dev: any, cumulative: any = True)\ - Log-normal distribution PDF or CDF.
- \lognorm_inv(probability: any, mean: any, standard_dev: any)\ - Inverse of the Log-normal cumulative distribution.
- \negbinom_dist(number_f: any, number_s: any, probability_s: any, cumulative: any = True)\ - Negative binomial distribution PMF or CDF.
- \norm_dist(x: any, mean: any = 0.0, std_dev: any = 1.0, cumulative: any = True)\ - Normal (Gaussian) distribution PDF or CDF.
- \norm_inv(probability: any, mean: any = 0.0, std_dev: any = 1.0)\ - Inverse of the normal cumulative distribution (quantile function).
- \norm_s_dist(z: any, cumulative: any = True)\ - Standard normal distribution (mean=0, std=1).
- \norm_s_inv(probability: any)\ - Inverse of the standard normal cumulative distribution.
- \phi(x: any)\ - Standard normal probability density function φ(x).
- \poisson_dist(x: any, mean: any, cumulative: any = True)\ - Poisson distribution PMF or CDF.
- \t_dist(x: any, deg_freedom: any, cumulative: any = True)\ - Student's t-distribution PDF or CDF.
- \t_dist_2t(x: any, deg_freedom: any)\ - Two-tailed Student's t-distribution.
- \t_dist_rt(x: any, deg_freedom: any)\ - Right-tailed Student's t-distribution.
- \t_inv(probability: any, deg_freedom: any)\ - Inverse of the Student's t cumulative distribution.
- \t_inv_2t(probability: any, deg_freedom: any)\ - Inverse of two-tailed Student's t-distribution.
- \weibull_dist(x: any, alpha: any, beta: any, cumulative: any = True)\ - Weibull distribution PDF or CDF.

## fxNumeric.finance_functions

- \accounts_payable_turnover(total_purchases: integer, average_payables: integer)\ - Calculate accounts payable turnover ratio.
- \accrint(issue: string, first_interest: string, settlement: string, rate: number, par: number, frequency: integer, basis: integer = 0)\ - Accrued interest for a security that pays periodic interest.
- \accrintm(issue: string, settlement: string, rate: number, par: number, basis: integer = 0)\ - Accrued interest for a security that pays interest at maturity.
- \acid_test_ratio(cash: integer, marketable_securities: integer, receivables: integer, current_liabilities: integer)\ - Calculate the acid-test (quick) ratio.
- \altman_z_score(working_capital: number, retained_earnings: number, ebit: number, market_cap: number, total_liabilities: number, total_assets: number, sales: number)\ - Calculates the Altman Z-Score for bankruptcy prediction.
- \amorlinc(cost: number, date_purchased: string, first_period: string, salvage: number, period: integer, rate: number, basis: integer = 0)\ - Linear depreciation for each accounting period (French system).
- \annuity_due_certain(payment: number, interest_rate: number, periods: integer)\ - Present value of an annuity-due (payments at beginning of each period).
- \annuity_due_fv(payment: number, rate: number, periods: integer)\ - Future value of an annuity due (payments at the beginning of each period).
- \annuity_due_pv(payment: number, rate: number, periods: integer)\ - Present value of an annuity due (payments at the beginning of each period).
- \annuity_immediate_certain(payment: number, interest_rate: number, periods: integer)\ - Present value of an annuity-immediate (payments at end of each period).
- \asset_coverage_ratio(total_assets: integer, intangible_assets: integer, current_liabilities: integer, total_debt: integer)\ - Return the asset coverage ratio.
- \asset_turnover(net_sales: integer, total_assets: integer)\ - Calculate the asset turnover ratio.
- \binomial_option_price(s: number, k: number, t: number, r: number, sigma: number, steps: integer, option_type: string = call)\ - European option price via the Cox-Ross-Rubinstein binomial tree.
- \black_scholes_call(s: number, k: number, t: number, r: number, sigma: number)\ - Calculates the price of a European call option (Black-Scholes model).
- \black_scholes_put(s: number, k: number, t: number, r: number, sigma: number)\ - Calculates the price of a European put option (Black-Scholes model).
- \bond_convexity(coupon_rate: number, ytm: number, periods: integer, face: number = 100.0)\ - Calculates the convexity of a bond.
- \bond_current_yield(annual_coupon: number, market_price: number)\ - Calculate the current yield of a bond.
- \bond_equivalent_yield(face_value: number, purchase_price: number, days_to_maturity: integer)\ - Bond equivalent yield (BEY) for a discount instrument.
- \bond_yield_to_worst(face: number, coupon_rate: number, price: number, years_to_maturity: integer, call_years: array = None)\ - Estimate the yield-to-worst for a callable bond.
- \break_even_units(fixed_costs: number, price_per_unit: number, variable_cost_per_unit: number)\ - Calculates the break-even point in units.
- \breakeven_units(fixed_costs: integer, price_per_unit: integer, variable_cost_per_unit: integer)\ - Calculate the break-even point in units.
- \burn_rate(starting_cash: integer, ending_cash: integer, months: integer)\ - Calculate the monthly cash burn rate.
- \cagr(beginning: number, ending: number, years: number)\ - Calculates the Compound Annual Growth Rate.
- \calmar_ratio(annualized_return: integer, max_drawdown: integer)\ - Calculate the Calmar ratio: annualized return / max drawdown.
- \capital_intensity_ratio(total_assets: integer, total_revenue: integer)\ - Return the capital intensity ratio.
- \capm(risk_free_rate: number, beta: number, market_return: number)\ - Computes the expected return using the Capital Asset Pricing Model.
- \cash_conversion_cycle(days_inventory: integer, days_receivable: integer, days_payable: integer)\ - Calculate the Cash Conversion Cycle: DIO + DSO - DPO.
- \cash_conversion_efficiency(operating_cash_flow: integer, net_income: integer)\ - Return the cash conversion efficiency ratio.
- \cash_flow_coverage(operating_cash_flow: integer, total_debt: integer)\ - Return the cash flow coverage ratio.
- \cash_ratio(cash: integer, current_liabilities: integer)\ - Calculate the cash ratio.
- \cash_return_on_assets(operating_cash_flow: integer, total_assets: integer)\ - Return the cash return on assets ratio.
- \compound_interest(principal: number, rate: number, n: integer, t: number)\ - Calculates compound interest (total amount minus principal).
- \conditional_var(returns: array, confidence: number = 0.95)\ - Conditional Value at Risk (CVaR / Expected Shortfall).
- \continuous_compounding(principal: number, rate: number, time: number)\ - Future value with continuous compounding.
- \contribution_margin(revenue: integer, variable_costs: integer)\ - Calculate the contribution margin: Revenue - Variable Costs.
- \convexity_adjustment(convexity: number, delta_yield: number)\ - Calculate the convexity adjustment for bond price change.
- \cost_of_debt_after_tax(interest_rate: number, tax_rate: number)\ - Calculate the after-tax cost of debt.
- \cost_of_equity_capm(risk_free_rate: integer, beta: integer, market_return: integer)\ - Calculate cost of equity using CAPM: Ke = Rf + β * (Rm - Rf).
- \coupdaybs(settlement: string, maturity: string, frequency: integer, basis: integer = 0)\ - Days from beginning of coupon period to settlement.
- \coupdays(settlement: string, maturity: string, frequency: integer, basis: integer = 0)\ - Number of days in the coupon period containing settlement.
- \coupdaysnc(settlement: string, maturity: string, frequency: integer, basis: integer = 0)\ - Days from settlement to next coupon date.
- \coupncd(settlement: string, maturity: string, frequency: integer)\ - Next coupon date after settlement.
- \coupnum(settlement: string, maturity: string, frequency: integer)\ - Number of coupons between settlement and maturity.
- \couppcd(settlement: string, maturity: string, frequency: integer)\ - Previous coupon date before settlement.
- \credit_spread(corporate_yield: number, risk_free_yield: number)\ - Credit spread between a corporate bond and risk-free rate.
- \cumipmt(rate: number, nper: integer, pv: number, start_period: integer, end_period: integer, type: integer = 0)\ - Calculates cumulative interest paid between two periods.
- \cumprinc(rate: number, nper: integer, pv: number, start_period: integer, end_period: integer, type: integer = 0)\ - Calculates cumulative principal paid between two periods.
- \current_ratio(current_assets: integer, current_liabilities: integer)\ - Computes the current (liquidity) ratio.
- \days_payable_outstanding(accounts_payable: integer, cost_of_goods_sold: integer, days: integer = 365)\ - Calculate days payable outstanding (DPO).
- \db(cost: number, salvage: number, life: integer, period: integer, month: integer = 12)\ - Calculates the depreciation of an asset for a specified period using the
- \ddb(cost: number, salvage: number, life: integer, period: integer, factor: number = 2.0)\ - Calculates depreciation using the double-declining-balance method.
- \debt_service_coverage(net_operating_income: integer, total_debt_service: integer)\ - Calculate the Debt Service Coverage Ratio (DSCR).
- \debt_service_ratio(net_operating_income: integer, total_debt_service: integer)\ - Calculate debt service coverage ratio (DSCR).
- \debt_to_capital(total_debt: integer, total_equity: integer)\ - Return the debt-to-capital ratio.
- \debt_to_equity(total_debt: number, total_equity: number)\ - Computes the debt-to-equity ratio.
- \debt_to_income(monthly_debt: number, monthly_income: number)\ - Calculate Debt-to-Income (DTI) ratio.
- \decreasing_annuity_pv(interest_rate: number, periods: integer)\ - Present value of a decreasing annuity-immediate paying (n−k+1) at time k.
- \defensive_interval(liquid_assets: integer, daily_operating_expenses: integer)\ - Return the defensive interval ratio (in days).
- \deferred_annuity_pv(payment: number, interest_rate: number, deferral: integer, periods: integer)\ - Present value of a deferred annuity-immediate.
- \degree_of_financial_leverage(ebit: integer, interest_expense: integer)\ - Calculate the degree of financial leverage (DFL).
- \disc(settlement: string, maturity: string, pr: number, redemption: number, basis: integer = 0)\ - Discount rate for a security.
- \discount_factor(rate: number, periods: integer)\ - Discount factor for a given rate and number of periods.
- \dividend_discount_price(dividend: integer, discount_rate: integer, periods: integer, growth_rate: integer = 0.0)\ - Calculate stock price via a finite-period dividend discount model.
- \dividend_payout_ratio(dividends_paid: integer, net_income: integer)\ - Calculate the dividend payout ratio as a percentage.
- \dividend_yield(annual_dividend: integer, market_price: integer)\ - Calculate the dividend yield as a percentage.
- \dollarde(fractional_dollar: number, fraction: integer)\ - Convert fractional dollar price to decimal.
- \dollarfr(decimal_dollar: number, fraction: integer)\ - Convert decimal dollar price to fractional notation.
- \dupont_roe(net_margin: integer, asset_turnover_ratio: integer, equity_multiplier: integer)\ - Calculate Return on Equity using the DuPont decomposition.
- \duration(settlement: string, maturity: string, coupon: number, yld: number, frequency: integer, basis: integer = 0)\ - Macaulay duration for a bond with par value $100.
- \duration_gap(asset_duration: number, liability_duration: number, leverage_ratio: number)\ - Duration gap for an institution's balance sheet.
- \earnings_per_share(net_income: integer, shares_outstanding: integer)\ - Computes earnings per share (EPS).
- \earnings_yield(earnings_per_share: integer, price_per_share: integer)\ - Calculate the earnings yield: EPS / Price (inverse of P/E ratio).
- \economic_order_quantity(annual_demand: number, order_cost: number, holding_cost: number)\ - Economic Order Quantity (EOQ) — Wilson's formula.
- \economic_value_added(nopat: integer, capital_employed: integer, wacc: integer)\ - Return the Economic Value Added (EVA).
- \effect(nominal_rate: number, npery: integer)\ - Calculates the effective annual interest rate.
- \effective_annual_rate(nominal_rate: integer, periods_per_year: integer)\ - Convert a nominal interest rate to the effective annual rate.
- \effective_duration(price_down: number, price_up: number, initial_price: number, delta_yield: number)\ - Calculate the effective duration of a bond.
- \endowment_insurance_pv(benefit: number, interest_rate: number, mortality_rates: array)\ - Present value of an endowment insurance.
- \enterprise_value_simple(market_cap_value: integer, total_debt: integer, cash: integer)\ - Calculate simplified enterprise value: EV = Market Cap + Debt - Cash.
- \equity_multiplier(total_assets: integer, total_equity: integer)\ - Calculate the equity multiplier (total assets / shareholders' equity).
- \ev_to_ebitda(enterprise_value: integer, ebitda: integer)\ - Calculate the Enterprise Value to EBITDA ratio.
- \expected_shortfall(portfolio_value: number, mean_return: number, std_return: number, confidence: number = 0.95)\ - Calculates parametric Expected Shortfall (CVaR) assuming normal returns.
- \fixed_asset_turnover(net_sales: integer, net_fixed_assets: integer)\ - Calculate fixed asset turnover ratio.
- \fixed_charge_coverage(ebit: integer, fixed_charges: integer)\ - Calculate the Fixed Charge Coverage Ratio.
- \force_of_interest(annual_rate: number)\ - Convert annual effective interest rate to force of interest (continuous rate).
- \force_of_mortality(mortality_rate: number)\ - Convert a one-year mortality probability to force of mortality.
- \free_cash_flow_yield(free_cash_flow: integer, market_cap: integer)\ - Return the free cash flow yield.
- \future_value(rate: number, nper: integer, pmt: number, pv: number, type: integer = 0)\ - Calculates the Future Value (FV) of an investment based on a series of
- \fvschedule(principal: number, schedule: array)\ - Calculates future value with a variable schedule of interest rates.
- \garman_klass_volatility(ohlc: array)\ - Garman-Klass volatility estimator from OHLC bars.
- \gordon_growth_price(dividend: integer, discount_rate: integer, growth_rate: integer)\ - Estimate the intrinsic price of a stock using the Gordon Growth Model.
- \gross_margin(revenue: integer, cogs: integer)\ - Computes the gross margin ratio.
- \gross_profit_margin(revenue: integer, cost_of_goods_sold: integer)\ - Calculate gross profit margin as a percentage.
- \growing_annuity_pv(payment: number, rate: number, growth: number, periods: integer)\ - Present value of a growing annuity.
- \growing_perpetuity_pv(payment: number, rate: number, growth: number)\ - Present value of a growing perpetuity.
- \historical_var(returns: array, confidence: number = 0.95)\ - Historical Value at Risk (VaR) at a given confidence level.
- \holding_period_return(initial_value: number, final_value: number, income: number = 0.0)\ - Holding period return (HPR).
- \implied_volatility(option_price: number, s: number, k: number, t: number, r: number, option_type: string = call, tol: number = 1e-06, max_iter: integer = 100)\ - Estimates implied volatility via bisection (standalone, no scipy).
- \increasing_annuity_pv(interest_rate: number, periods: integer)\ - Present value of an increasing annuity-immediate paying k at time k.
- \inflation_adjusted_value(amount: integer, rate: integer, years: integer)\ - Calculate the inflation-adjusted (real) value of money.
- \information_ratio(portfolio_return: integer, benchmark_return: integer, tracking_error: integer)\ - Calculate the Information Ratio: (Rp - Rb) / TE.
- \interest_burden_ratio(pre_tax_income: integer, ebit: integer)\ - Return the interest burden ratio (DuPont component).
- \interest_coverage_ratio(ebit: integer, interest_expense: integer)\ - Calculate the interest coverage ratio.
- \intrate(settlement: string, maturity: string, investment: number, redemption: number, basis: integer = 0)\ - Interest rate for a fully invested security.
- \inventory_to_sales(inventory: integer, net_sales: integer)\ - Return the inventory-to-sales ratio.
- \inventory_turnover(cogs: integer, average_inventory: integer)\ - Calculate the inventory turnover ratio.
- \inventory_turnover_ratio(cost_of_goods_sold: integer, average_inventory: integer)\ - Calculate inventory turnover ratio.
- \ipmt(rate: number, per: integer, nper: integer, pv: number, fv: number = 0.0, type_: integer = 0)\ - Calculates the interest portion of a payment for a given period.
- \irr(cash_flows: array, guess: number = 0.1)\ - Calculates the Internal Rate of Return (IRR) for a series of cash flows.
- \ispmt(rate: number, per: integer, nper: integer, pv: number)\ - Interest paid during a specific period of an investment.
- \kelly_criterion(win_probability: integer, win_loss_ratio: integer)\ - Calculate the Kelly criterion: f* = p - (1-p)/b.
- \leverage_ratio(total_assets: integer, total_equity: integer)\ - Calculate the leverage ratio (equity multiplier): Assets / Equity.
- \levered_beta(asset_beta: number, tax_rate: number, de_ratio: number)\ - Calculate levered (equity) beta using the inverse Hamada equation.
- \life_annuity_pv(annual_payment: number, interest_rate: number, mortality_rates: array)\ - Present value of a life annuity-due using discrete mortality rates.
- \loan_amortization_table(principal: number, annual_rate: number, periods: integer)\ - Generates a loan amortization table with fixed payments.
- \loan_to_value(loan_amount: number, appraised_value: number)\ - Calculate the Loan-to-Value (LTV) ratio.
- \macaulay_duration_simple(coupon_rate: number, ytm: number, periods: integer)\ - Calculates Macaulay duration from coupon rate and yield to maturity.
- \margin_of_safety_pct(actual_sales: integer, breakeven_sales: integer)\ - Calculate the margin of safety as a percentage.
- \marginal_cost(change_in_total_cost: integer, change_in_quantity: integer)\ - Return the marginal cost of production.
- \markdown_percentage(original_price: integer, sale_price: integer)\ - Calculate markdown (discount) percentage.
- \market_cap(share_price: integer, shares_outstanding: integer)\ - Calculate market capitalisation: Price × Shares Outstanding.
- \markup_percentage(cost: integer, selling_price: integer)\ - Calculate markup percentage (margin over cost).
- \mduration(settlement: string, maturity: string, coupon: number, yld: number, frequency: integer, basis: integer = 0)\ - Modified duration for a bond with par value $100.
- \mirr(cash_flows: array, finance_rate: number, reinvest_rate: number)\ - Calculates the Modified Internal Rate of Return.
- \modified_dietz_return(start_value: number, end_value: number, cash_flows: array, cf_days: array, total_days: integer)\ - Calculate the Modified Dietz rate of return.
- \modified_duration_simple(mac_duration: number, ytm: number, frequency: integer = 1)\ - Calculates modified duration from Macaulay duration.
- \money_weighted_return(cashflows: array, guess: number = 0.05, tolerance: number = 1e-09, max_iterations: integer = 200)\ - Money-weighted rate of return (IRR of investor cash flows).
- \months_of_runway(cash_available: integer, monthly_burn_rate: integer)\ - Calculate financial runway in months.
- \mortgage_remaining_balance(principal: number, annual_rate: number, years: integer, payments_made: integer)\ - Calculates the remaining balance of a fixed-rate mortgage after N payments.
- \mortgage_total_cost(principal: number, annual_rate: number, years: integer)\ - Calculates the total cost of a fixed-rate mortgage (principal + interest).
- \net_asset_value(total_assets: number, total_liabilities: number, shares: number)\ - Calculate Net Asset Value (NAV) per share.
- \net_profit_margin(net_income: integer, revenue: integer)\ - Calculate the net profit margin as a percentage.
- \nominal(effect_rate: number, npery: integer)\ - Calculates the nominal annual interest rate from effective rate.
- \nominal_to_real_rate(nominal_rate: integer, inflation_rate: integer)\ - Convert nominal interest rate to real rate using the Fisher equation.
- \nopat_margin(nopat: integer, revenue: integer)\ - Return the NOPAT margin.
- \nper(rate: number, pmt: number, pv: number, fv: number = 0.0, type: integer = 0)\ - Calculates the number of periods for an investment based on periodic,
- \npv(rate: number, values: array)\ - Calculates the Net Present Value (NPV) of an investment.
- \omega_ratio(returns: array, threshold: number = 0.0)\ - Calculate the Omega ratio of a return series.
- \operating_cash_flow_ratio(operating_cash_flow: integer, current_liabilities: integer)\ - Calculate the Operating Cash Flow Ratio.
- \operating_expense_ratio(operating_expenses: integer, net_sales: integer)\ - Calculate the operating expense ratio (OER).
- \operating_leverage_degree(contribution_margin_value: integer, operating_income: integer)\ - Calculate the Degree of Operating Leverage: DOL = CM / EBIT.
- \operating_margin(operating_income: integer, revenue: integer)\ - Computes the operating margin ratio.
- \operating_profit_margin(operating_income: integer, revenue: integer)\ - Return the operating profit margin.
- \option_charm_bs(s: number, k: number, t: number, r: number, sigma: number, option_type: string = call)\ - Black-Scholes charm (delta decay): rate of change of delta w.r.t. time.
- \option_color_bs(s: number, k: number, t: number, r: number, sigma: number)\ - Black-Scholes color: rate of change of gamma w.r.t. time (∂Γ/∂t).
- \option_delta_bs(s: number, k: number, t: number, r: number, sigma: number, option_type: string = call)\ - Black-Scholes delta for a European option.
- \option_gamma_bs(s: number, k: number, t: number, r: number, sigma: number)\ - Black-Scholes gamma for a European option.
- \option_rho_bs(s: number, k: number, t: number, r: number, sigma: number, option_type: string = call)\ - Black-Scholes rho: sensitivity to the risk-free interest rate.
- \option_speed_bs(s: number, k: number, t: number, r: number, sigma: number)\ - Black-Scholes speed: rate of change of gamma w.r.t. spot price (∂³C/∂S³).
- \option_theta_bs(s: number, k: number, t: number, r: number, sigma: number, option_type: string = call)\ - Black-Scholes theta: time decay of an option per year.
- \option_vanna_bs(s: number, k: number, t: number, r: number, sigma: number)\ - Black-Scholes vanna: sensitivity of delta to volatility (∂²C/∂S∂σ).
- \option_vega_bs(s: number, k: number, t: number, r: number, sigma: number)\ - Black-Scholes vega: sensitivity of option price to volatility.
- \option_zomma_bs(s: number, k: number, t: number, r: number, sigma: number)\ - Black-Scholes zomma: rate of change of gamma w.r.t. volatility.
- \parkinson_volatility(high_low: array)\ - Parkinson volatility estimator from high-low prices.
- \payback_period(initial_investment: number, cash_flows: array)\ - Calculates the simple payback period of an investment.
- \pduration(rate: number, pv: number, fv: number)\ - Calculates the number of periods to reach a target future value.
- \plowback_ratio(dividends_per_share: integer, earnings_per_share: integer)\ - Calculate the plowback (retention) ratio from per-share values.
- \pmt(rate: number, nper: integer, pv: number, fv: number = 0.0, type: integer = 0)\ - Calculates the payment for a loan based on constant payments and a
- \ppmt(rate: number, per: integer, nper: integer, pv: number, fv: number = 0.0, type_: integer = 0)\ - Calculates the principal portion of a payment for a given period.
- \present_value(rate: number, nper: integer, pmt: number, fv: number = 0.0, type: integer = 0)\ - Calculates the Present Value (PV) of an investment based on a series of
- \present_value_perpetuity(payment: number, rate: number)\ - Present value of a perpetuity (infinite stream of equal payments).
- \price(settlement: string, maturity: string, rate: number, yld: number, redemption: number, frequency: integer, basis: integer = 0)\ - Price per $100 face of a security paying periodic interest.
- \price_earnings_growth(pe_ratio: integer, earnings_growth_rate: integer)\ - Return the Price/Earnings-to-Growth (PEG) ratio.
- \price_to_book(market_price: integer, book_value_per_share: integer)\ - Calculate the Price-to-Book (P/B) ratio.
- \price_to_earnings(market_price: integer, earnings_per_share_value: integer)\ - Calculate the Price-to-Earnings (P/E) ratio.
- \price_to_sales(market_cap: integer, total_revenue: integer)\ - Calculate the price-to-sales (P/S) ratio.
- \pricedisc(settlement: string, maturity: string, discount: number, redemption: number, basis: integer = 0)\ - Price per $100 face of a discounted security.
- \pricemat(settlement: string, maturity: string, issue: string, rate: number, yld: number, basis: integer = 0)\ - Price per $100 face of a security that pays at maturity.
- \profitability_index(rate: number, initial_investment: number, cash_flows: array)\ - Calculates the Profitability Index (PI) of a project.
- \purchasing_power(amount: integer, inflation_rate: integer, years: integer)\ - Calculate remaining purchasing power after inflation erodes value.
- \put_call_parity_check(call_price: number, put_price: number, s: number, k: number, r: number, t: number, tolerance: number = 0.01)\ - Check if put-call parity holds within a tolerance.
- \quick_ratio(current_assets: integer, inventories: integer, current_liabilities: integer)\ - Calculate the quick (acid-test) ratio.
- \rate(nper: integer, pmt: number, pv: number, fv: number = 0.0, type: integer = 0, guess: number = 0.1)\ - Calculates the interest rate per period of an annuity.
- \real_rate_of_return(nominal_rate: integer, inflation_rate: integer)\ - Computes the real rate of return using the Fisher equation.
- \receivables_turnover(net_credit_sales: integer, average_receivables: integer)\ - Calculate accounts receivable turnover ratio.
- \received(settlement: string, maturity: string, investment: number, discount: number, basis: integer = 0)\ - Amount received at maturity for a fully invested security.
- \retention_ratio(net_income: integer, dividends_paid: integer)\ - Calculate the retention ratio (plowback ratio).
- \return_on_assets(net_income: integer, total_assets: integer)\ - Computes return on assets (ROA).
- \return_on_capital_employed(ebit: integer, total_assets: integer, current_liabilities: integer)\ - Calculate return on capital employed (ROCE).
- \return_on_equity(net_income: integer, shareholders_equity: integer)\ - Computes return on equity (ROE).
- \return_on_net_assets(net_income: integer, fixed_assets: integer, working_capital: integer)\ - Calculate return on net assets (RONA).
- \revenue_per_employee(total_revenue: integer, number_of_employees: integer)\ - Calculate revenue generated per employee.
- \roi(gain: number, cost: number)\ - Calculates Return on Investment as a percentage.
- \rri(nper: integer, pv: number, fv: number)\ - Equivalent interest rate for the growth of an investment.
- \rule_of_72(rate: integer)\ - Estimates the number of periods to double an investment.
- \sales_growth_rate(current_sales: integer, previous_sales: integer)\ - Return the sales growth rate as a decimal.
- \sharpe_ratio(returns: array, risk_free_rate: number = 0.0)\ - Computes the Sharpe ratio of a return series.
- \simple_interest(principal: number, rate: number, time_periods: number)\ - Calculates simple interest.
- \sln(cost: number, salvage: number, life: integer)\ - Calculates the straight-line depreciation of an asset for one period.
- \sortino_ratio(returns: array, risk_free_rate: number = 0.0)\ - Computes the Sortino ratio (uses downside deviation only).
- \spot_to_forward(spot1: number, t1: number, spot2: number, t2: number)\ - Implied forward rate between two spot rates.
- \sustainable_growth_rate(roe: integer, retention: integer)\ - Calculate the sustainable growth rate: SGR = ROE × Retention Ratio.
- \syd(cost: number, salvage: number, life: integer, per: integer)\ - Calculates depreciation using the sum-of-years' digits method.
- \tbilleq(settlement: string, maturity: string, discount: number)\ - Bond-equivalent yield for a Treasury bill.
- \tbillprice(settlement: string, maturity: string, discount: number)\ - Price per $100 face for a Treasury bill.
- \tbillyield(settlement: string, maturity: string, pr: number)\ - Yield for a Treasury bill.
- \term_life_insurance_pv(benefit: number, interest_rate: number, mortality_rates: array)\ - Present value of a term life insurance (death benefit paid at end of year of death).
- \time_weighted_return(period_returns: array)\ - Time-weighted rate of return (geometric linking of sub-period returns).
- \times_interest_earned(ebit: integer, interest_expense: integer)\ - Calculate times interest earned (TIE) ratio.
- \tobin_q_ratio(market_value: integer, replacement_cost: integer)\ - Return Tobin's Q ratio.
- \treynor_ratio(portfolio_return: integer, risk_free_rate: integer, beta: integer)\ - Calculate the Treynor ratio: (Rp - Rf) / β.
- \ulcer_index(values: array)\ - Calculate the Ulcer Index for a price or value series.
- \unlevered_beta(levered_beta: number, tax_rate: number, de_ratio: number)\ - Calculate unlevered (asset) beta using the Hamada equation.
- \var_parametric(portfolio_value: number, mean_return: number, std_return: number, confidence: number = 0.95)\ - Calculates parametric Value at Risk (VaR) assuming normal returns.
- \vdb(cost: number, salvage: number, life: number, start_period: number, end_period: number, factor: number = 2.0, no_switch: boolean = False)\ - Variable declining balance depreciation.
- \wacc(equity: number, debt: number, cost_of_equity: number, cost_of_debt: number, tax_rate: number)\ - Computes the Weighted Average Cost of Capital.
- \wacc_two_sources(equity: integer, cost_of_equity: integer, debt: integer, cost_of_debt: integer, tax_rate: integer)\ - Return the Weighted Average Cost of Capital (two-source model).
- \weighted_average_cost_of_debt(debts: array, rates: array, tax_rate: number = 0.0)\ - Compute the weighted average after-tax cost of debt.
- \working_capital(current_assets: integer, current_liabilities: integer)\ - Calculate working capital.
- \working_capital_ratio(current_assets: integer, current_liabilities: integer)\ - Calculate the working capital ratio (current ratio).
- \xirr(values: array, dates: array, guess: number = 0.1)\ - Calculates Internal Rate of Return for irregularly spaced cash flows.
- \xnpv(rate: number, values: array, dates: array)\ - Calculates Net Present Value for irregularly spaced cash flows.
- \yang_zhang_volatility(ohlc: array)\ - Yang-Zhang volatility estimator from OHLC bars.
- \yield_bond(settlement: string, maturity: string, rate: number, pr: number, redemption: number, frequency: integer, basis: integer = 0)\ - Yield of a security that pays periodic interest.
- \yielddisc(settlement: string, maturity: string, pr: number, redemption: number, basis: integer = 0)\ - Yield for a discounted security.
- \yieldmat(settlement: string, maturity: string, issue: string, rate: number, pr: number, basis: integer = 0)\ - Yield of a security that pays interest at maturity.
- \zero_coupon_price(face_value: number, yield_rate: number, years_to_maturity: number)\ - Price of a zero-coupon bond.

## fxNumeric.finite_differences_functions

- \backward_difference_table(y: array)\ - Builds a backward-difference table ∇ⁿy.
- \bessel_interpolation(x_values: array, y_values: array, x: number)\ - Bessel's central-difference interpolation formula.
- \central_difference_table(y: array)\ - Builds a central-difference table δⁿy.
- \divided_difference_table(x_values: array, y_values: array)\ - Builds a divided-difference table for Newton's general interpolation.
- \forward_difference_table(y: array)\ - Builds a forward-difference table Δⁿy.
- \gauss_forward_interpolation(x_values: array, y_values: array, x: number)\ - Gauss's forward central-difference interpolation formula.
- \newton_backward_interpolation(x_values: array, y_values: array, x: number)\ - Newton's backward-difference interpolation formula.
- \newton_forward_interpolation(x_values: array, y_values: array, x: number)\ - Newton's forward-difference interpolation formula.
- \stirling_interpolation(x_values: array, y_values: array, x: number)\ - Stirling's central-difference interpolation formula.

## fxNumeric.format_functions

- \compare_floats_with_tolerance(a: number, b: number, rel_tol: number = 1e-09, abs_tol: number = 1e-05)\ - Compares two floating-point numbers to determine if they are "close" to each other.
- \format_as_percentage(number: number, decimals: integer = 2)\ - Formats a floating-point number as a percentage string.
- \format_as_scientific_notation(number: number, decimals: integer = 2)\ - Formats a floating-point number in scientific notation.
- \format_with_leading_zeros(number: integer, width: integer)\ - Formats an integer as a string, padding with leading zeros to a specific width.
- \percent_change(old_value: number, new_value: number)\ - Calculates the percentage change between two values.
- \percent_of(part: number, whole: number)\ - Calculates what percentage ``part`` is of ``whole``.

## fxNumeric.geometry_functions

- \annulus_area(outer_radius: number, inner_radius: number)\ - Compute the area of an annulus (ring).
- \arc_length(radius: number, angle: number)\ - Length of a circular arc: r * theta.
- \capsule_surface_area(radius: number, cylinder_length: number)\ - Compute the surface area of a capsule.
- \capsule_volume(radius: number, cylinder_length: number)\ - Compute the volume of a capsule (cylinder with hemispherical caps).
- \circle_area(radius: number)\ - Area of a circle: pi * r^2.
- \circle_circumference(radius: number)\ - Circumference of a circle: 2 * pi * r.
- \circular_ring_perimeter(outer_radius: number, inner_radius: number)\ - Compute the perimeter of an annulus (both circles).
- \cone_lateral_area(radius: number, slant_height: number)\ - Lateral surface area of a cone: pi r l.
- \cone_slant_height(radius: number, height: number)\ - Compute the slant height of a cone.
- \cone_volume(radius: number, height: number)\ - Volume of a right circular cone: (1/3) pi r^2 h.
- \cuboid_space_diagonal(a: number, b: number, c: number)\ - Compute the space diagonal of a cuboid.
- \cylinder_surface_area(radius: number, height: number)\ - Total surface area of a right circular cylinder: 2 pi r (r + h).
- \cylinder_volume(radius: number, height: number)\ - Volume of a right circular cylinder: pi r^2 h.
- \distance_2d(x1: number, y1: number, x2: number, y2: number)\ - Euclidean distance between two points in the plane.
- \distance_3d(x1: number, y1: number, z1: number, x2: number, y2: number, z2: number)\ - Euclidean distance between two points in 3-D space.
- \dodecahedron_surface_area(edge: number)\ - Compute the surface area of a regular dodecahedron.
- \dodecahedron_volume(edge: number)\ - Compute the volume of a regular dodecahedron.
- \ellipse_area(a: number, b: number)\ - Area of an ellipse: pi * a * b.
- \ellipse_eccentricity(a: number, b: number)\ - Eccentricity of an ellipse: e = sqrt(1 - (b/a)^2) where a >= b.
- \ellipse_perimeter_approx(a: number, b: number)\ - Approximate perimeter of an ellipse using Ramanujan's second formula.
- \ellipsoid_volume(a: number, b: number, c: number)\ - Volume of an ellipsoid: (4/3) pi a b c.
- \frustum_volume(r1: number, r2: number, height: number)\ - Volume of a frustum (truncated cone): (pi h / 3)(r1^2 + r1*r2 + r2^2).
- \haversine_distance(lat1: number, lon1: number, lat2: number, lon2: number, radius: number = 6371.0)\ - Great-circle distance between two points on a sphere (Haversine formula).
- \heron_formula(a: number, b: number, c: number)\ - Area of a triangle from side lengths using Heron's formula.
- \hyperbola_asymptotes(a: number, b: number)\ - Slopes of the asymptotes of x^2/a^2 - y^2/b^2 = 1.
- \hyperbola_eccentricity(a: number, b: number)\ - Eccentricity of a hyperbola: e = sqrt(1 + (b/a)^2).
- \icosahedron_surface_area(edge: number)\ - Compute the surface area of a regular icosahedron.
- \icosahedron_volume(edge: number)\ - Compute the volume of a regular icosahedron.
- \law_of_cosines_angle(a: number, b: number, c: number)\ - Finds angle C opposite to side c using the law of cosines.
- \law_of_cosines_side(a: number, b: number, angle_c: number)\ - Finds side c using the law of cosines: c^2 = a^2 + b^2 - 2ab cos(C).
- \law_of_sines_side(a: number, angle_a: number, angle_b: number)\ - Finds side b using the law of sines: b = a * sin(B) / sin(A).
- \line_equation(x1: number, y1: number, x2: number, y2: number)\ - Returns coefficients (a, b, c) of the line ax + by + c = 0 through two points.
- \midpoint_2d(x1: number, y1: number, x2: number, y2: number)\ - Midpoint of the segment connecting two 2-D points.
- \midpoint_3d(x1: number, y1: number, z1: number, x2: number, y2: number, z2: number)\ - Midpoint of the segment connecting two 3-D points.
- \octahedron_surface_area(edge: number)\ - Compute the surface area of a regular octahedron.
- \octahedron_volume(edge: number)\ - Compute the volume of a regular octahedron.
- \parabola_directrix(a: number)\ - Directrix of the parabola y = ax^2: y = -1/(4a).
- \parabola_focus(a: number)\ - Focus of the parabola y = ax^2: located at (0, 1/(4a)).
- \parallelogram_area(base: number, height: number)\ - Area of a parallelogram: base * height.
- \point_to_line_distance(px: number, py: number, a: number, b: number, c: number)\ - Distance from point (px, py) to the line ax + by + c = 0.
- \polygon_area(vertices: array)\ - Area of a simple polygon using the Shoelace formula.
- \polygon_centroid(vertices: array)\ - Compute the centroid of a simple polygon.
- \prism_surface_area(base_area: number, base_perimeter: number, height: number)\ - Compute the surface area of a prism.
- \prism_volume(base_area: number, height: number)\ - Compute the volume of a prism.
- \pyramid_surface_area(base_area: number, base_perimeter: number, slant_height: number)\ - Compute the surface area of a regular pyramid.
- \pyramid_volume(base_area: number, height: number)\ - Volume of a pyramid: (1/3) A_base h.
- \regular_polygon_area(n_sides: integer, side: number)\ - Area of a regular n-sided polygon: (n s^2) / (4 tan(pi/n)).
- \regular_polygon_interior_angle(n: integer)\ - Compute the interior angle of a regular n-gon in degrees.
- \regular_polygon_perimeter(n: integer, side: number)\ - Compute the perimeter of a regular polygon.
- \rhombus_area(d1: number, d2: number)\ - Compute the area of a rhombus from its diagonals.
- \rhombus_perimeter(side: number)\ - Compute the perimeter of a rhombus.
- \sector_area(radius: number, angle: number)\ - Area of a circular sector: (1/2) r^2 theta.
- \segment_area(radius: number, angle: number)\ - Area of a circular segment: (r^2/2)(theta - sin(theta)).
- \shoelace_area(vertices: array)\ - Compute the area of a simple polygon using the shoelace formula.
- \slope_two_points(x1: number, y1: number, x2: number, y2: number)\ - Slope of the line through two points (y2-y1)/(x2-x1).
- \sphere_surface_area(radius: number)\ - Surface area of a sphere: 4 pi r^2.
- \sphere_volume(radius: number)\ - Volume of a sphere: (4/3) pi r^3.
- \spherical_cap_surface_area(radius: number, height: number)\ - Compute the curved surface area of a spherical cap.
- \spherical_cap_volume(radius: number, h: number)\ - Volume of a spherical cap: (pi h^2 / 3)(3R - h).
- \spherical_excess(a: number, b: number, c: number)\ - Spherical excess of a spherical triangle (L'Huilier's theorem).
- \spherical_law_of_cosines(a: number, b: number, angle_c: number)\ - Spherical law of cosines: cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(C).
- \stadium_area(radius: number, straight_length: number)\ - Compute the area of a stadium (discorectangle).
- \tetrahedron_surface_area(edge: number)\ - Compute the surface area of a regular tetrahedron.
- \tetrahedron_volume(edge: number)\ - Compute the volume of a regular tetrahedron.
- \torus_surface_area(major_r: number, minor_r: number)\ - Surface area of a torus: 4 pi^2 R r.
- \torus_volume(major_r: number, minor_r: number)\ - Volume of a torus: 2 pi^2 R r^2.
- \trapezoid_area(a: number, b: number, h: number)\ - Area of a trapezoid: (a + b) * h / 2.
- \triangle_area_vertices(x1: number, y1: number, x2: number, y2: number, x3: number, y3: number)\ - Area of a triangle from its vertices using the Shoelace formula.

## fxNumeric.inequalities_functions

- \am_gm_inequality(values: array)\ - Verifies the AM-GM inequality: AM >= GM for positive values.
- \am_hm_inequality(values: array)\ - Verifies the AM-HM inequality: AM >= HM for positive values.
- \bernoulli_inequality(x: number, n: integer)\ - Verifies Bernoulli's inequality: (1+x)^n >= 1 + nx for x >= -1, n >= 1.
- \cauchy_schwarz_inequality(a: array, b: array)\ - Verifies the Cauchy-Schwarz inequality: (Σ a_i b_i)² <= (Σ a_i²)(Σ b_i²).
- \chebyshev_sum_inequality(a: array, b: array)\ - Verifies Chebyshev's sum inequality for similarly ordered sequences.
- \holder_inequality(a: array, b: array, p: number = 2.0)\ - Verifies Hölder's inequality: Σ|a_i b_i| <= ||a||_p · ||b||_q where 1/p + 1/q = 1.
- \jensen_inequality(f: number, values: array, weights: array = None, convex: boolean = True)\ - Verifies Jensen's inequality for convex/concave functions.
- \minkowski_inequality(a: array, b: array, p: number = 2.0)\ - Verifies Minkowski's inequality: ||a+b||_p <= ||a||_p + ||b||_p.
- \power_mean(values: array, p: number)\ - Computes the generalized power mean M_p of positive values.
- \rearrangement_inequality(a: array, b: array)\ - Computes the rearrangement inequality bounds.
- \triangle_inequality_vector(v: array)\ - Verifies the triangle inequality for a vector: |Σ v_i| <= Σ |v_i|.
- \young_inequality(a: number, b: number, p: number = 2.0)\ - Verifies Young's inequality: ab <= a^p/p + b^q/q where 1/p + 1/q = 1.

## fxNumeric.interpolation_functions

- \bilinear_interpolation(x: integer, y: integer, x1: integer, x2: integer, y1: integer, y2: integer, q11: integer, q21: integer, q12: integer, q22: integer)\ - Performs bilinear interpolation on a 2D grid.
- \clamp_interpolate(x: number, x0: number, y0: number, x1: number, y1: number)\ - Linearly interpolate between two points, clamped to the output range.
- \clip_number(x: integer, min_val: integer, max_val: integer)\ - Forces a number to be within a specific range [min_val, max_val].
- \cubic_spline_natural(x_points: array, y_points: array, x: integer)\ - Evaluates a natural cubic spline at x.
- \inverse_interpolate(y: number, y0: number, y1: number)\ - Compute the interpolation factor t such that lerp(y0, y1, t) == y.
- \lagrange_interpolation(x_points: array, y_points: array, x: integer)\ - Evaluates the Lagrange interpolating polynomial at a given x.
- \linear_interpolate(x: number, x0: number, y0: number, x1: number, y1: number)\ - Linearly interpolate between two points.
- \linear_interpolation(x: integer, x_points: array, y_points: array)\ - Estimates a Y value for *x* via piecewise linear interpolation.
- \map_range(value: integer, in_min: integer, in_max: integer, out_min: integer, out_max: integer)\ - Re-map a value from one range to another (Arduino-style).
- \newton_divided_difference(x_points: array, y_points: array, x: integer)\ - Evaluates a polynomial at x using Newton's divided-difference form.
- \normalize_list(data: array)\ - Normalizes a list to the [0, 1] range using min-max scaling.
- \normalize_to_0_1_range(x: integer, min_val: integer, max_val: integer)\ - Normalizes a number to scale it to a range between 0 and 1.
- \piecewise_interpolate(x: number, xs: array, ys: array)\ - Linearly interpolate over a piecewise-defined dataset.
- \scale_to_new_range(x: integer, min_x: integer, max_x: integer, new_min: integer, new_max: integer)\ - Scales a number from an original range to a new range.

## fxNumeric.mechanics_functions

- \centroid_circular_sector(r: number, alpha: number)\ - Returns centroid distance from center of a circular sector.
- \centroid_cone(h: number)\ - Returns the height of the centroid of a solid cone above its base.
- \centroid_hemisphere(r: number)\ - Returns the height of the centroid of a solid hemisphere above the flat face.
- \centroid_polygon(vertices: array)\ - Computes the centroid of a simple polygon using the Shoelace-derived formula.
- \centroid_semicircle(r: number)\ - Returns the centroid of a semicircle of radius r lying above the x-axis.
- \centroid_triangle(p1: number, p2: number, p3: number)\ - Computes the centroid of a triangle given its vertices.
- \mass_moment_cone(m: number, r: number)\ - Mass moment of inertia of a solid cone about its axis.
- \mass_moment_cylinder(m: number, r: number)\ - Mass moment of inertia of a solid cylinder about its axis.
- \mass_moment_cylinder_transverse(m: number, r: number, h: number)\ - Mass moment of inertia of a solid cylinder about a transverse centroidal axis.
- \mass_moment_disk(m: number, r: number)\ - Mass moment of inertia of a thin uniform disk about its axis.
- \mass_moment_hollow_sphere(m: number, r: number)\ - Mass moment of inertia of a thin hollow sphere about any diameter.
- \mass_moment_rod(m: number, length: number)\ - Mass moment of inertia of a thin rod about its center (perpendicular axis).
- \mass_moment_sphere(m: number, r: number)\ - Mass moment of inertia of a solid sphere about any diameter.
- \moment_of_inertia_annulus(r_outer: number, r_inner: number)\ - Computes the area moment of inertia of an annulus (hollow circle) about a diameter.
- \moment_of_inertia_circle(r: number)\ - Computes the area moment of inertia of a circle about a diameter.
- \moment_of_inertia_ellipse(a: number, b: number)\ - Computes the area moments of inertia of an ellipse about its centroidal axes.
- \moment_of_inertia_rectangle(b: number, h: number)\ - Computes the area moments of inertia of a rectangle about centroidal axes.
- \moment_of_inertia_semicircle(r: number)\ - Computes the area moment of inertia of a semicircle about its diameter.
- \moment_of_inertia_triangle(b: number, h: number)\ - Computes the area moment of inertia of a triangle about its base.
- \moment_of_inertia_triangle_centroidal(b: number, h: number)\ - Computes the area moment of inertia of a triangle about its centroidal axis.
- \parallel_axis_theorem(i_cm: number, m: number, d: number)\ - Applies the parallel axis theorem: I = I_cm + md².
- \polar_moment_of_inertia(ix: number, iy: number)\ - Computes the polar moment of inertia J = I_x + I_y.
- \radius_of_gyration(i: number, m: number)\ - Computes the radius of gyration k = sqrt(I/m).
- \section_modulus(i: number, c: number)\ - Computes the section modulus S = I/c.

## fxNumeric.number_theory_functions

- \additive_persistence(n: integer)\ - Count the additive persistence of n.
- \aliquot_sum(n: integer)\ - Compute the aliquot sum s(n) = σ₁(n) - n (sum of proper divisors).
- \bell_number(n: integer)\ - Return the *n*-th Bell number.
- \big_omega(n: integer)\ - Count prime factors with multiplicity Ω(n).
- \carmichael_lambda(n: integer)\ - Compute the Carmichael function λ(n).
- \catalan_number(n: integer)\ - Calculates the nth Catalan number.
- \chinese_remainder_theorem(remainders: array, moduli: array)\ - Solves a system of simultaneous congruences via CRT.
- \collatz_length(n: integer)\ - Counts the number of steps in the Collatz sequence until reaching 1.
- \collatz_steps(n: integer)\ - Count steps to reach 1 in the Collatz sequence.
- \continued_fraction_expansion(numerator: integer, denominator: integer)\ - Compute the continued fraction expansion of a rational number.
- \convergents(cf: array)\ - Compute the convergents of a continued fraction.
- \coprime(a: integer, b: integer)\ - Check if a and b are coprime (gcd(a, b) = 1).
- \count_digits(n: integer)\ - Counts the number of decimal digits of an integer.
- \count_divisors(n: integer)\ - Count the number of positive divisors of n.
- \delannoy_number(m: integer, n: integer)\ - Return the central Delannoy number D(m, n).
- \digit_count(n: integer)\ - Count the number of digits in a non-negative integer.
- \digit_factorial_sum(n: integer)\ - Compute the sum of factorials of the digits of n.
- \digital_root(n: integer)\ - Calculate the digital root of n.
- \digital_sum(n: integer)\ - Computes the sum of digits of an integer.
- \discrete_logarithm(g: integer, h: integer, p: integer)\ - Compute the discrete logarithm: find *x* such that g^x ≡ h (mod p).
- \divisors(n: integer)\ - Returns all positive divisors of *n* in ascending order.
- \egyptian_fraction(numerator: integer, denominator: integer)\ - Decompose a fraction into a sum of distinct unit fractions.
- \euler_totient(n: integer)\ - Calculates Euler's totient function phi(n).
- \eulerian_number(n: integer, k: integer)\ - Return the Eulerian number A(n, k).
- \fibonacci_sequence(n: integer)\ - Returns the first *n* Fibonacci numbers.
- \get_primes_up_to(limit: integer)\ - Generates a list of prime numbers up to a given limit using the Sieve of Eratosthenes.
- \goldbach_partition(n: integer)\ - Finds a Goldbach partition for an even integer >= 4.
- \integer_partitions_count(n: integer)\ - Counts the number of integer partitions of n.
- \is_abundant_number(n: integer)\ - Check if n is an abundant number (σ₁(n) > 2n).
- \is_achilles_number(n: integer)\ - Check if n is an Achilles number.
- \is_amicable_pair(a: integer, b: integer)\ - Check if (a, b) is an amicable pair: s(a) = b and s(b) = a.
- \is_automorphic_number(n: integer)\ - Check if n is automorphic: n² ends with n in decimal representation.
- \is_coprime(a: integer, b: integer)\ - Checks whether two integers are coprime.
- \is_cube_number(n: integer)\ - Check if n is a perfect cube.
- \is_deficient_number(n: integer)\ - Check if n is a deficient number (σ₁(n) < 2n).
- \is_even(n: integer)\ - Checks whether an integer is even.
- \is_fibonacci_number(n: integer)\ - Check whether a non-negative integer belongs to the Fibonacci sequence.
- \is_happy_number(n: integer)\ - Check if n is a happy number.
- \is_harshad_number(n: integer)\ - Check if n is a Harshad (Niven) number (divisible by its digit sum).
- \is_kaprekar_number(n: integer)\ - Check if n is a Kaprekar number.
- \is_lychrel_candidate(n: integer, max_iterations: integer = 50)\ - Check if a number is a Lychrel candidate.
- \is_narcissistic_number(n: integer)\ - Checks if n is a narcissistic (Armstrong) number.
- \is_odd(n: integer)\ - Checks whether an integer is odd.
- \is_palindrome_number(n: integer)\ - Checks whether an integer is a palindrome.
- \is_palindromic_number(n: integer)\ - Check if n is a palindromic number in base 10.
- \is_perfect_number(n: integer)\ - Check if n is a perfect number (σ₁(n) = 2n).
- \is_perfect_power(n: integer)\ - Check if n can be expressed as a^b where a, b are integers and b > 1.
- \is_power_of_two(n: integer)\ - Checks whether a positive integer is a power of two.
- \is_powerful_number(n: integer)\ - Check if n is a powerful number (every prime factor appears ≥ 2 times).
- \is_prime(n: integer)\ - Checks if a number is prime.
- \is_pronic_number(n: integer)\ - Check if n is a pronic (oblong) number: n = k * (k + 1) for some k >= 0.
- \is_semiprime(n: integer)\ - Check if n is a semiprime (product of exactly two primes).
- \is_smith_number(n: integer)\ - Check if n is a Smith number.
- \is_sphenic_number(n: integer)\ - Check if n is a sphenic number (product of exactly 3 distinct primes).
- \is_square_free(n: integer)\ - Check if n is square-free (no prime factor appears more than once).
- \is_square_number(n: integer)\ - Checks whether a non-negative integer is a perfect square.
- \is_triangular_number(n: integer)\ - Checks whether a non-negative integer is a triangular number.
- \jacobi_symbol(a: integer, n: integer)\ - Compute the Jacobi symbol (a/n).
- \jacobsthal_number(n: integer)\ - Calculate the n-th Jacobsthal number.
- \kronecker_symbol(a: integer, n: integer)\ - Compute the Kronecker symbol (a|n).
- \legendre_symbol(a: integer, p: integer)\ - Compute the Legendre symbol (a/p) for odd prime p.
- \liouville_lambda(n: integer)\ - Compute the Liouville function λ(n) = (-1)^Ω(n).
- \lucas_number(n: integer)\ - Calculate the n-th Lucas number.
- \lucas_sequence(n: integer)\ - Generates the first n Lucas numbers.
- \mobius_function(n: integer)\ - Compute the Möbius function μ(n).
- \modular_exponentiation(base: integer, exponent: integer, modulus: integer)\ - Computes (base ** exponent) % modulus efficiently.
- \modular_inverse(a: integer, m: integer)\ - Computes the modular multiplicative inverse of a modulo m.
- \motzkin_number(n: integer)\ - Return the *n*-th Motzkin number.
- \multiplicative_persistence(n: integer)\ - Count the multiplicative persistence of n.
- \narayana_number(n: integer, k: integer)\ - Return the Narayana number N(n, k).
- \next_prime(n: integer)\ - Finds the smallest prime number greater than n.
- \nth_fibonacci(n: integer)\ - Returns the n-th Fibonacci number using fast doubling.
- \number_of_distinct_prime_factors(n: integer)\ - Count the number of distinct prime factors of n (ω(n)).
- \number_of_divisors(n: integer)\ - Compute d(n) = σ₀(n), the number of divisors of n.
- \omega_prime(n: integer)\ - Count distinct prime factors ω(n).
- \partition_function(n: integer)\ - Computes the number of integer partitions p(n).
- \partition_number(n: integer)\ - Return the number of integer partitions of *n*.
- \pell_number(n: integer)\ - Calculate the n-th Pell number.
- \pentagonal_number(n: integer)\ - Return the *n*-th pentagonal number.
- \prime_counting_approx(x: number)\ - Approximate π(x), the number of primes ≤ x, using x/ln(x).
- \prime_factors(n: integer)\ - Returns the prime factorization of *n* in ascending order.
- \prime_nth_approx(n: integer)\ - Approximate the n-th prime using n·ln(n) + n·ln(ln(n)).
- \primitive_root(p: integer)\ - Finds the smallest primitive root modulo p (p must be prime).
- \radical(n: integer)\ - Compute the radical of n: product of distinct prime factors.
- \schroeder_number(n: integer)\ - Return the *n*-th large Schröder number.
- \squarefree_part(n: integer)\ - Computes the squarefree part of n.
- \stern_brocot(n: integer)\ - Generates the first n elements of the Stern-Brocot sequence.
- \stirling_number_second(n: integer, k: integer)\ - Return the Stirling number of the second kind S(n, k).
- \sum_of_cubes_of_digits(n: integer)\ - Calculate the sum of cubes of the digits of n.
- \sum_of_divisors(n: integer, k: integer = 1)\ - Compute σ_k(n), the sum of k-th powers of divisors of n.
- \sum_of_squares_count(n: integer)\ - Counts the number of representations of n as a sum of two squares.
- \sum_of_squares_of_digits(n: integer)\ - Calculate the sum of squares of the digits of n.
- \sum_proper_divisors(n: integer)\ - Calculate the sum of proper divisors of n (all divisors except n itself).
- \tribonacci_number(n: integer)\ - Calculate the n-th tribonacci number.

## fxNumeric.numerical_methods_functions

- \adaptive_rk45(f: number, y0: number, t_start: number, t_end: number, tol: number = 1e-08, h_init: number = 0.01, h_min: number = 1e-12, h_max: number = 1.0)\ - Solves the ODE y' = f(t, y) using the adaptive Runge-Kutta-Fehlberg (RK45) method.
- \backward_difference(f: number, x: number, h: number = 1e-07)\ - Approximates f'(x) using the backward difference formula.
- \bisection_method(f: number, a: number, b: number, tol: number = 1e-12, max_iter: integer = 1000)\ - Finds a root of f in [a, b] using the bisection method.
- \central_difference(f: number, x: number, h: number = 1e-07)\ - Approximates f'(x) using the central difference formula.
- \euler_method(f: number, y0: number, t_start: number, t_end: number, n: integer = 1000)\ - Solves the ODE y' = f(t, y) using Euler's forward method.
- \fixed_point_iteration(g: number, x0: number, tol: number = 1e-12, max_iter: integer = 1000)\ - Finds a fixed point x = g(x) via simple iteration.
- \forward_difference(f: number, x: number, h: number = 1e-07)\ - Approximates f'(x) using the forward difference formula.
- \gaussian_quadrature(f: number, a: number, b: number, n: integer = 5)\ - Approximates the definite integral of f from a to b using Gauss-Legendre quadrature.
- \midpoint_rule(f: number, a: number, b: number, n: integer = 1000)\ - Approximates the definite integral of f from a to b using the midpoint rule.
- \monte_carlo_integration(f: number, a: number, b: number, n: integer = 100000, seed: integer = None)\ - Approximates the definite integral of f from a to b via Monte Carlo sampling.
- \newton_raphson(f: number, df: number, x0: number, tol: number = 1e-12, max_iter: integer = 1000)\ - Finds a root of f using Newton-Raphson iteration.
- \ode_system_rk4(f: number, y0: array, t_start: number, t_end: number, n: integer = 1000)\ - Solves a system of ODEs y' = f(t, y) using the 4th-order Runge-Kutta method.
- \regula_falsi(f: number, a: number, b: number, tol: number = 1e-12, max_iter: integer = 1000)\ - Finds a root of f in [a, b] using the false position (regula falsi) method.
- \richardson_extrapolation(f: number, x: number, h: number = 0.1, order: integer = 4)\ - Approximates f'(x) using Richardson extrapolation on central differences.
- \romberg_integration(f: number, a: number, b: number, max_order: integer = 10, tol: number = 1e-12)\ - Approximates the definite integral of f from a to b using Romberg's method.
- \runge_kutta_2(f: number, y0: number, t_start: number, t_end: number, n: integer = 1000)\ - Solves the ODE y' = f(t, y) using the 2nd-order Runge-Kutta (midpoint) method.
- \runge_kutta_4(f: number, y0: number, t_start: number, t_end: number, n: integer = 1000)\ - Solves the ODE y' = f(t, y) using the classic 4th-order Runge-Kutta method.
- \secant_method(f: number, x0: number, x1: number, tol: number = 1e-12, max_iter: integer = 1000)\ - Finds a root of f using the secant method.
- \second_derivative_central(f: number, x: number, h: number = 1e-05)\ - Approximates f''(x) using the central difference formula for second derivatives.
- \simpsons_38_rule(f: number, a: number, b: number, n: integer = 999)\ - Approximates the definite integral of f from a to b using Simpson's 3/8 rule.
- \simpsons_rule(f: number, a: number, b: number, n: integer = 1000)\ - Approximates the definite integral of f from a to b using Simpson's 1/3 rule.
- \trapezoidal_rule(f: number, a: number, b: number, n: integer = 1000)\ - Approximates the definite integral of f from a to b using the trapezoidal rule.

## fxNumeric.polynomial_functions

- \partial_fraction_simple(numerator: array, denominator_roots: array)\ - Partial-fraction coefficients for simple (non-repeated) linear factors.
- \polynomial_add(p: array, q: array)\ - Adds two polynomials.
- \polynomial_composition(p: array, q: array)\ - Computes the composition p(q(x)).
- \polynomial_degree(p: array)\ - Returns the degree of a polynomial.
- \polynomial_derivative(p: array)\ - Computes the derivative of a polynomial.
- \polynomial_divide(p: array, q: array)\ - Divides polynomial p by q, returning (quotient, remainder).
- \polynomial_evaluate(p: array, x: integer)\ - Evaluates a polynomial at x using Horner's method.
- \polynomial_from_roots(roots: array)\ - Constructs a monic polynomial from its roots.
- \polynomial_gcd(p: array, q: array)\ - Computes the GCD of two polynomials using the Euclidean algorithm.
- \polynomial_integral(p: array, constant: integer = 0)\ - Computes the indefinite integral of a polynomial.
- \polynomial_multiply(p: array, q: array)\ - Multiplies two polynomials.
- \polynomial_scale(p: array, scalar: integer)\ - Scales a polynomial by a scalar factor.
- \polynomial_subtract(p: array, q: array)\ - Subtracts polynomial q from p.

## fxNumeric.probability_extra_functions

- \bernoulli_mean(p: number)\ - Mean of the Bernoulli distribution: E[X] = p.
- \bernoulli_pmf(k: integer, p: number)\ - Probability mass function of the Bernoulli distribution.
- \bernoulli_variance(p: number)\ - Variance of the Bernoulli distribution: Var(X) = p(1-p).
- \cauchy_cdf(x: number, x0: number = 0.0, gamma: number = 1.0)\ - Cumulative distribution function of the Cauchy distribution.
- \cauchy_pdf(x: number, x0: number = 0.0, gamma: number = 1.0)\ - Probability density function of the Cauchy distribution.
- \cauchy_quantile(p: number, x0: number = 0.0, gamma: number = 1.0)\ - Quantile (inverse CDF) of the Cauchy distribution.
- \geometric_cdf(k: integer, p: number)\ - Cumulative distribution function of the geometric distribution.
- \geometric_mean_dist(p: number)\ - Mean of the geometric distribution: E[X] = 1/p.
- \geometric_pmf(k: integer, p: number)\ - Probability mass function of the geometric distribution.
- \geometric_variance_dist(p: number)\ - Variance of the geometric distribution: Var(X) = (1-p)/p².
- \log_logistic_cdf(x: number, alpha: number, beta: number)\ - CDF of the log-logistic distribution.
- \log_logistic_pdf(x: number, alpha: number, beta: number)\ - PDF of the log-logistic distribution.
- \maxwell_boltzmann_cdf(x: number, a: number = 1.0)\ - CDF of the Maxwell-Boltzmann distribution.
- \maxwell_boltzmann_mean(a: number = 1.0)\ - Mean of the Maxwell-Boltzmann distribution: 2a√(2/π).
- \maxwell_boltzmann_pdf(x: number, a: number = 1.0)\ - PDF of the Maxwell-Boltzmann distribution.
- \pareto_cdf(x: number, x_m: number, alpha: number)\ - CDF of the Pareto distribution.
- \pareto_mean(x_m: number, alpha: number)\ - Mean of the Pareto distribution: α x_m / (α-1) for α > 1.
- \pareto_pdf(x: number, x_m: number, alpha: number)\ - PDF of the Pareto distribution.
- \pareto_variance(x_m: number, alpha: number)\ - Variance of the Pareto distribution: x_m² α / ((α-1)²(α-2)) for α > 2.
- \rayleigh_cdf(x: number, sigma: number = 1.0)\ - CDF of the Rayleigh distribution.
- \rayleigh_mean(sigma: number = 1.0)\ - Mean of the Rayleigh distribution: σ√(π/2).
- \rayleigh_pdf(x: number, sigma: number = 1.0)\ - PDF of the Rayleigh distribution.
- \rayleigh_variance(sigma: number = 1.0)\ - Variance of the Rayleigh distribution: (4-π)/2 σ².
- \uniform_continuous_cdf(x: number, a: number, b: number)\ - CDF of the continuous uniform distribution U(a, b).
- \uniform_continuous_mean(a: number, b: number)\ - Mean of U(a, b): (a+b)/2.
- \uniform_continuous_pdf(x: number, a: number, b: number)\ - PDF of the continuous uniform distribution U(a, b).
- \uniform_continuous_variance(a: number, b: number)\ - Variance of U(a, b): (b-a)²/12.
- \uniform_discrete_cdf(k: integer, a: integer, b: integer)\ - CDF of the discrete uniform distribution.
- \uniform_discrete_pmf(k: integer, a: integer, b: integer)\ - PMF of the discrete uniform distribution on {a, a+1, ..., b}.

## fxNumeric.random_functions

- \random_array(rows: integer = 1, columns: integer = 1, min_value: number = 0.0, max_value: number = 1.0, whole_number: boolean = False)\ - Returns an array of random numbers.
- \random_bool(probability: number = 0.5)\ - Return a random boolean with the given probability of True.
- \random_choice(items: any)\ - Return a random element from a non-empty sequence.
- \random_float(low: number = 0.0, high: number = 1.0)\ - Return a random float in the range [low, high).
- \random_gaussian(mean: number = 0.0, std: number = 1.0)\ - Return a random float from a Gaussian (normal) distribution.
- \random_int(low: integer, high: integer)\ - Return a random integer in the inclusive range [low, high].
- \random_sample(items: any, k: integer)\ - Return k unique elements chosen from the sequence without replacement.
- \random_shuffle(items: any)\ - Return a new list with the elements randomly shuffled.
- \random_uuid()\ - Generate a random UUID version 4 as a string.
- \random_weighted_choice(items: any, weights: number)\ - Return a random element using weighted probabilities.
- \set_random_seed(seed: integer)\ - Set the random seed for reproducible generation.

## fxNumeric.rounding_functions

- \add_with_exact_precision(num1: number, num2: number)\ - Adds two numbers using the Decimal type to avoid precision errors.
- \ceiling_math(number: number, significance: number = 1.0, mode: integer = 0)\ - Round *number* up to the nearest multiple of *significance*.
- \ceiling_precise(number: number, significance: number = 1.0)\ - Round *number* up to the nearest multiple of *significance*.
- \ceiling_significance(number: number, significance: number = 1.0)\ - Rounds a number up to the nearest multiple of significance.
- \even(number: number)\ - Rounds a number up to the nearest even integer.
- \floor_math(number: number, significance: number = 1.0, mode: integer = 0)\ - Round *number* down to the nearest multiple of *significance*.
- \floor_precise(number: number, significance: number = 1.0)\ - Round *number* down to the nearest multiple of *significance*.
- \floor_significance(number: number, significance: number = 1.0)\ - Rounds a number down to the nearest multiple of significance.
- \manual_round_and_cast(number: number)\ - Performs manual rounding to the nearest integer and then casts to int.
- \manual_round_down_to_int(number: number)\ - Rounds a floating-point number to the nearest integer, always downwards.
- \manual_round_up_to_int(number: number)\ - Rounds a floating-point number to the nearest integer, always upwards.
- \odd(number: number)\ - Rounds a number up to the nearest odd integer.
- \quantize_number(x: integer, step: integer)\ - Quantizes a number, forcing it to take only certain discrete values (multiples of 'step').
- \round_down(number: number)\ - Rounds a floating-point number always down to the nearest integer.
- \round_half_even(number: number, target_precision: string = 1)\ - Performs banker's rounding (ROUND_HALF_EVEN) of a number to a specific precision.
- \round_to_n_decimals(number: number, decimals: integer)\ - Rounds a floating-point number to a specific number of decimals.
- \round_to_nearest_multiple(number: integer, base: integer)\ - Rounds a number to the nearest multiple of a given base.
- \round_up(number: number)\ - Rounds a floating-point number always up to the nearest integer.
- \truncate_float(number: number)\ - Truncates a floating-point number, removing the decimal part.

## fxNumeric.series_functions

- \abel_sum(series: array, r: number = 0.99)\ - Compute the Abel sum of a series: Σ a_n r^n.
- \alternating_series_sum(f: integer, n: integer)\ - Computes an alternating series sum_{k=0}^{n-1} (-1)^k f(k).
- \arithmetic_series_sum(a: number, d: number, n: integer)\ - Computes the sum of an arithmetic series: a + (a+d) + ... + (a+(n-1)d).
- \basel_series(terms: integer = 1000)\ - Approximates pi^2/6 using the Basel series: sum 1/k^2.
- \bbp_pi_digit(n: integer)\ - Compute π using the Bailey-Borwein-Plouffe formula (partial sum).
- \binomial_series(alpha: number, x: number, terms: integer = 20)\ - Evaluates the generalized binomial series (1+x)^alpha for |x| < 1.
- \cesaro_sum(series: array)\ - Compute the Cesàro sum of a series.
- \chudnovsky_pi(terms: integer = 5)\ - Approximate π using the Chudnovsky algorithm.
- \convergent_sequence(f: any, x0: number, terms: integer = 100)\ - Generate a convergent sequence by repeated application x_{n+1} = f(x_n).
- \euler_maclaurin_sum(f: any, a: integer, b: integer, terms: integer = 4)\ - Approximate Σ_{k=a}^{b} f(k) using the Euler-Maclaurin formula.
- \fourier_coefficients(f: number, period: number, n_terms: integer, n_points: integer = 1000)\ - Computes numerical Fourier series coefficients a_n and b_n.
- \fourier_series_eval(a_coeffs: array, b_coeffs: array, period: number, x: number)\ - Evaluates a Fourier series at point x given its coefficients.
- \generalized_harmonic(n: integer, s: number)\ - Computes the generalized harmonic number H_{n,s} = sum_{k=1}^{n} 1/k^s.
- \geometric_series_infinite(a: number, r: number)\ - Computes the sum of an infinite geometric series: a / (1 - r).
- \geometric_series_sum(a: number, r: number, n: integer)\ - Computes the sum of a finite geometric series: a + ar + ar^2 + ... + ar^{n-1}.
- \gregory_leibniz_pi(terms: integer = 1000)\ - Approximate π using the Gregory-Leibniz series.
- \harmonic_series_partial(n: integer)\ - Computes the n-th partial sum of the harmonic series H_n = sum_{k=1}^{n} 1/k.
- \leibniz_pi(terms: integer = 1000)\ - Approximates pi/4 using the Leibniz formula: sum (-1)^k / (2k+1).
- \madhava_pi(terms: integer = 50)\ - Approximate π using Madhava's series (predecessor to Gregory-Leibniz).
- \nilakantha_pi(terms: integer = 1000)\ - Approximate π using the Nilakantha series.
- \p_series_partial(n: integer, p: number)\ - Computes the partial sum of the p-series sum_{k=1}^{n} 1/k^p.
- \power_series_eval(coefficients: array, x: number)\ - Evaluates a power series sum_{k=0}^{n} a_k x^k using Horner's method.
- \ramanujan_pi(terms: integer = 10)\ - Approximate π using Ramanujan's series.
- \taylor_asin(x: number, terms: integer = 50)\ - Approximates arcsin(x) via the Maclaurin series for |x| <= 1.
- \taylor_atan(x: number, terms: integer = 50)\ - Compute arctan(x) via Taylor/Maclaurin series.
- \taylor_atanh(x: number, terms: integer = 50)\ - Compute arctanh(x) via Taylor series for |x| < 1.
- \taylor_cos(x: number, terms: integer = 20)\ - Approximates cos(x) using the Maclaurin series.
- \taylor_cosh(x: number, terms: integer = 30)\ - Compute cosh(x) via Taylor series.
- \taylor_exp(x: number, terms: integer = 50)\ - Compute e^x via Taylor series.
- \taylor_ln1p(x: number, terms: integer = 50)\ - Approximates ln(1+x) using the Maclaurin series for |x| <= 1.
- \taylor_log1p(x: number, terms: integer = 100)\ - Compute ln(1+x) via Taylor series for |x| ≤ 1.
- \taylor_sin(x: number, terms: integer = 20)\ - Approximates sin(x) using the Maclaurin series.
- \taylor_sinh(x: number, terms: integer = 30)\ - Compute sinh(x) via Taylor series.
- \vieta_pi(terms: integer = 30)\ - Approximate π using Vieta's infinite product.
- \wallis_pi(terms: integer = 1000)\ - Approximate π using Wallis' product.
- \zeta_even(n: integer)\ - Compute ζ(2n) = (-1)^{n+1} B_{2n} (2π)^{2n} / (2(2n)!).

## fxNumeric.special_functions

- \abel_polynomial(n: integer, x: number, a: number = 1.0)\ - Compute the n-th Abel polynomial p_n(x; a) = x(x - na)^{n-1}.
- \airy_ai(x: number)\ - Computes the Airy function Ai(x) via power-series/asymptotic expansion.
- \airy_bi(x: number)\ - Computes the Airy function Bi(x) via power-series expansion.
- \anger_j(nu: number, x: number, n_points: integer = 1000)\ - Anger function J_ν(x) = (1/π) ∫_0^π cos(νθ - x sinθ) dθ.
- \associated_legendre(n: integer, m: integer, x: number)\ - Evaluates the associated Legendre function P_n^m(x).
- \bell_number(n: integer)\ - Compute the n-th Bell number using the Bell triangle.
- \bernoulli_number(n: integer)\ - Computes the n-th Bernoulli number B_n using the Akiyama-Tanigawa algorithm.
- \bernoulli_polynomial(n: integer, x: number)\ - Evaluates the n-th Bernoulli polynomial B_n(x).
- \bernstein_polynomial(n: integer, k: integer, t: number)\ - Evaluate the Bernstein basis polynomial B_{n,k}(t).
- \bessel_j(n: integer, x: number, terms: integer = 80)\ - Computes the Bessel function of the first kind J_n(x) via power series.
- \bessel_y(n: integer, x: number, terms: integer = 80)\ - Computes the Bessel function of the second kind Y_n(x) (Neumann function).
- \beta_function(a: number, b: number)\ - Computes the beta function B(a, b) = Gamma(a)*Gamma(b)/Gamma(a+b).
- \catalan_number(n: integer)\ - Compute the n-th Catalan number C_n = (2n)! / ((n+1)! · n!).
- \central_binomial_coefficient(n: integer)\ - Compute the central binomial coefficient C(2n, n).
- \chebyshev_polynomial_first(n: integer, x: number)\ - Evaluates the Chebyshev polynomial of the first kind T_n(x).
- \chebyshev_polynomial_second(n: integer, x: number)\ - Evaluates the Chebyshev polynomial of the second kind U_n(x).
- \clausen_function(x: number, terms: integer = 100)\ - Compute the Clausen function Cl₂(x) = -∫₀ˣ ln|2 sin(t/2)| dt.
- \complementary_error_function(x: number)\ - Compute the complementary error function erfc(x) = 1 - erf(x).
- \cosine_integral(x: number, terms: integer = 200)\ - Computes the cosine integral Ci(x) = γ + ln(x) + ∫₀ˣ (cos(t)-1)/t dt.
- \dawson_function(x: number, terms: integer = 100)\ - Computes Dawson's function D(x) = e^(-x^2) ∫₀ˣ e^(t^2) dt.
- \debye_function(n: integer, x: number, terms: integer = 200)\ - Compute the Debye function D_n(x) = (n/x^n) ∫₀ˣ t^n/(e^t - 1) dt.
- \dedekind_eta(tau_im: number, terms: integer = 100)\ - Computes the Dedekind eta function η(τ) for purely imaginary τ = i·τ_im.
- \digamma(x: number)\ - Computes the digamma (psi) function: derivative of ln(Gamma(x)).
- \dilogarithm(z: number)\ - Computes the dilogarithm (Spence's function) Li_2(z) for z <= 1.
- \dirichlet_integral()\ - Returns the exact value of the Dirichlet integral: integral_{0}^{inf} sin(x)/x dx = pi/2.
- \double_factorial(n: integer)\ - Compute the double factorial n!! = n × (n-2) × (n-4) × ··· × (1 or 2).
- \elliptic_e(k: number)\ - Computes the complete elliptic integral of the second kind E(k).
- \elliptic_e_incomplete(phi: number, k: number)\ - Computes the incomplete elliptic integral of the second kind E(phi, k).
- \elliptic_f(phi: number, k: number)\ - Computes the incomplete elliptic integral of the first kind F(phi, k).
- \elliptic_k(k: number)\ - Computes the complete elliptic integral of the first kind K(k).
- \elliptic_pi(n_param: number, k: number)\ - Computes the complete elliptic integral of the third kind Pi(n, k).
- \error_function(x: number)\ - Compute the error function erf(x) via Maclaurin series.
- \euler_number(n: integer)\ - Computes the n-th Euler number E_n.
- \euler_polynomial(n: integer, x: number)\ - Evaluates the n-th Euler polynomial E_n(x).
- \euler_reflection_formula(z: number)\ - Verifies Euler's reflection formula: Γ(z)Γ(1-z) = π/sin(πz).
- \exponential_integral_ei(x: number, terms: integer = 200)\ - Computes the exponential integral Ei(x) for x > 0.
- \falling_factorial(x: number, n: integer)\ - Computes the falling factorial x_(n) = x(x-1)(x-2)...(x-n+1).
- \fibonacci_polynomial(n: integer, x: number)\ - Compute the n-th Fibonacci polynomial F_n(x).
- \fresnel_c(x: number, terms: integer = 100)\ - Computes the Fresnel cosine integral C(x) = ∫₀ˣ cos(π t²/2) dt.
- \fresnel_s(x: number, terms: integer = 100)\ - Computes the Fresnel sine integral S(x) = ∫₀ˣ sin(π t²/2) dt.
- \gamma_duplication_formula(z: number)\ - Legendre's duplication formula: Γ(z)Γ(z+1/2) = √π · Γ(2z) / 2^(2z-1).
- \gaussian_integral()\ - Returns the exact value of the Gaussian integral: integral_{-inf}^{inf} e^{-x^2} dx = sqrt(pi).
- \gegenbauer_polynomial(n: integer, alpha: number, x: number)\ - Gegenbauer (ultraspherical) polynomial C_n^α(x) via recurrence.
- \generalized_harmonic_number(n: integer, m: number = 1.0)\ - Compute the generalized harmonic number H(n, m) = Σ_{k=1}^{n} 1/k^m.
- \generalized_laguerre_polynomial(n: integer, alpha: number, x: number)\ - Evaluates the generalized (associated) Laguerre polynomial L_n^alpha(x).
- \gudermannian(x: number)\ - Compute the Gudermannian function gd(x) = 2·arctan(tanh(x/2)).
- \harmonic_number(n: integer)\ - Compute the n-th harmonic number H_n = Σ_{k=1}^{n} 1/k.
- \heaviside_step(x: number)\ - Compute the Heaviside step function H(x).
- \hermite_polynomial(n: integer, x: number)\ - Evaluates the (physicist's) Hermite polynomial H_n(x).
- \hexagonal_number(n: integer)\ - Compute the n-th hexagonal number H_n = n(2n - 1).
- \hurwitz_zeta(s: number, a: number, terms: integer = 100000)\ - Computes the Hurwitz zeta function ζ(s, a) = Σ 1/(n+a)^s for n=0,1,2,...
- \hypergeometric_1f1(a: number, b: number, z: number)\ - Computes the confluent hypergeometric function 1F1(a; b; z) (Kummer's M).
- \hypergeometric_2f1(a: number, b: number, c: number, z: number)\ - Computes the Gauss hypergeometric function 2F1(a, b; c; z).
- \incomplete_beta_regularized(a: number, b: number, x: number)\ - Computes the regularized incomplete beta function I_x(a, b).
- \incomplete_gamma_lower(a: number, x: number)\ - Computes the regularized lower incomplete gamma function P(a, x).
- \inverse_error_function(p: number)\ - Compute the inverse error function erfinv(p) via Newton-Raphson.
- \inverse_gudermannian(x: number)\ - Compute the inverse Gudermannian gd⁻¹(x) = ln(tan(π/4 + x/2)).
- \jacobi_polynomial(n: integer, alpha: number, beta: number, x: number)\ - Jacobi polynomial P_n^(α,β)(x) via recurrence.
- \jacobi_theta_1(z: number, q: number, terms: integer = 50)\ - Computes the Jacobi theta function θ₁(z, q).
- \jacobi_theta_2(z: number, q: number, terms: integer = 50)\ - Computes the Jacobi theta function θ₂(z, q).
- \jacobi_theta_3(z: number, q: number, terms: integer = 50)\ - Computes the Jacobi theta function θ₃(z, q).
- \jacobi_theta_4(z: number, q: number, terms: integer = 50)\ - Computes the Jacobi theta function θ₄(z, q).
- \kelvin_bei(x: number, terms: integer = 50)\ - Kelvin function bei(x) = Im[J_0(x√i)].
- \kelvin_ber(x: number, terms: integer = 50)\ - Kelvin function ber(x) = Re[J_0(x√i)].
- \laguerre_polynomial(n: integer, x: number)\ - Evaluates the Laguerre polynomial L_n(x).
- \lambert_w(x: number, tol: number = 1e-12)\ - Compute the principal branch W₀ of the Lambert W function.
- \lanczos_gamma(z: number)\ - Computes the Gamma function using the Lanczos approximation.
- \legendre_polynomial(n: integer, x: number)\ - Evaluates the Legendre polynomial P_n(x) via Bonnet's recurrence.
- \log_gamma(x: number)\ - Compute the natural logarithm of the Gamma function ln(Γ(x)).
- \logistic_function(x: number, upper: number = 1.0, k: number = 1.0, x0: number = 0.0)\ - Compute the logistic (sigmoid) function f(x) = L / (1 + e^{-k(x - x₀)}).
- \lucas_polynomial(n: integer, x: number)\ - Compute the n-th Lucas polynomial L_n(x).
- \mittag_leffler(alpha: number, beta: number, z: number, terms: integer = 100)\ - Mittag-Leffler function E_{α,β}(z) = Σ_{k=0}^∞ z^k / Γ(αk+β).
- \modified_bessel_i(n: integer, x: number, terms: integer = 80)\ - Computes the modified Bessel function of the first kind I_n(x).
- \modified_bessel_k(n: integer, x: number)\ - Computes the modified Bessel function of the second kind K_n(x).
- \multinomial_coefficient(n: integer, groups: array = ...)\ - Computes the multinomial coefficient n! / (k1! * k2! * ... * km!).
- \normalized_sinc(x: number)\ - Compute the normalized sinc function sinc(x) = sin(πx)/(πx).
- \partition_number(n: integer)\ - Compute the partition number p(n) using dynamic programming.
- \pentagonal_number(n: integer)\ - Compute the n-th pentagonal number P_n = n(3n - 1)/2.
- \polygamma(m: integer, x: number)\ - Computes the polygamma function ψ^(m)(x) numerically.
- \polylogarithm(s: number, z: number, terms: integer = 500)\ - Computes the polylogarithm Li_s(z) = Σ z^k / k^s for |z| <= 1.
- \ramp_function(x: number)\ - Compute the ramp function R(x) = max(0, x).
- \reciprocal_gamma(x: number)\ - Reciprocal gamma function 1/Γ(x).
- \rectangular_function(x: number)\ - Compute the rectangular (boxcar) function rect(x).
- \riemann_zeta(s: number)\ - Computes the Riemann zeta function zeta(s) for real s > 1.
- \rising_factorial(x: number, n: integer)\ - Computes the rising factorial (Pochhammer symbol) x^(n) = x(x+1)(x+2)...(x+n-1).
- \sinc_function(x: number)\ - Compute the unnormalized sinc function sinc(x) = sin(x)/x.
- \sine_integral(x: number, terms: integer = 200)\ - Computes the sine integral Si(x) = ∫₀ˣ sin(t)/t dt.
- \softplus_function(x: number)\ - Compute the softplus function f(x) = ln(1 + e^x).
- \spherical_bessel_j(n: integer, x: number)\ - Computes the spherical Bessel function of the first kind j_n(x).
- \spherical_bessel_y(n: integer, x: number)\ - Computes the spherical Bessel function of the second kind y_n(x).
- \spherical_harmonic_real(l: integer, m: integer, theta: number, phi: number)\ - Computes the real spherical harmonic Y_l^m(theta, phi).
- \stirling_approximation(n: integer)\ - Computes Stirling's approximation for n!: sqrt(2πn) * (n/e)^n.
- \stirling_number_first(n: integer, k: integer)\ - Computes the unsigned Stirling number of the first kind |s(n, k)|.
- \struve_h(nu: integer, x: number, terms: integer = 50)\ - Struve function H_ν(x) for integer ν ≥ 0.
- \subfactorial(n: integer)\ - Computes the subfactorial !n (number of derangements of n elements).
- \touchard_polynomial(n: integer, x: number)\ - Compute the n-th Touchard (exponential) polynomial T_n(x).
- \triangular_function(x: number)\ - Compute the triangular function tri(x) = max(0, 1 - |x|).
- \trigamma(x: number)\ - Computes the trigamma function ψ₁(x) = d²/dx² ln(Γ(x)).
- \upper_incomplete_gamma(a: number, x: number, terms: integer = 200)\ - Computes the upper incomplete gamma function Γ(a, x) = ∫ₓ^∞ t^(a-1) e^(-t) dt.
- \wallis_product(n: integer)\ - Computes the Wallis product approximation for pi/2 using n terms.
- \weierstrass_p(z: number, omega1: number = 1.0, omega3_im: number = 1.0, terms: integer = 10)\ - Computes the Weierstrass ℘-function for real z with rectangular lattice.

## fxNumeric.statistics_functions

- \absolute_error(measured: integer, actual: integer)\ - Compute the absolute error: |measured - actual|.
- \adjusted_r_squared(r2: number, n: integer, p: integer)\ - Adjusted R² penalised for number of predictors.
- \aggregate(data: array, operation: string = sum, ignore_errors: boolean = False)\ - Perform a named aggregation on a numeric list.
- \anderson_darling(data: array)\ - Compute the Anderson-Darling normality test statistic (standalone).
- \argmax(data: array)\ - Returns the index of the maximum value in a list.
- \argmin(data: array)\ - Returns the index of the minimum value in a list.
- \auto_correlation(data: array, lag: integer = 1)\ - Calculates the autocorrelation of a time series at a given lag.
- \autocorrelation(data: array, lag: integer = 1)\ - Sample autocorrelation at a given lag.
- \average_deviation(data: array)\ - Calculates the average of absolute deviations from the mean.
- \average_if(values: array, criteria_range: array, criteria: any)\ - Return the average of *values* where the parallel *criteria_range* meets *criteria*.
- \balanced_accuracy_scalar(tp: integer, tn: integer, fp: integer, fn: integer)\ - Calculate balanced accuracy: (Sensitivity + Specificity) / 2.
- \bayes_posterior(prior: number, likelihood: number, evidence: number)\ - Compute the Bayesian posterior probability.
- \bayes_theorem(prior: number, likelihood: number, evidence: number)\ - Posterior probability via Bayes' theorem.
- \bayesian_update(prior: number, likelihoods: array, marginals: array)\ - Sequential Bayesian update with multiple evidence events.
- \benford_distribution(data: array)\ - Compute the leading-digit frequency distribution for Benford's law analysis.
- \beta_function(a: number, b: number)\ - Compute the beta function B(a, b) = Γ(a)Γ(b)/Γ(a+b).
- \beta_function_value(a: integer, b: integer)\ - Return the value of the Beta function B(a, b).
- \bfill(data: array)\ - Backward-fills None values with the next non-None value.
- \binary_entropy(p: integer)\ - Compute binary (Bernoulli) entropy: -p*log2(p) - (1-p)*log2(1-p).
- \binomial_pmf(k: integer, n: integer, p: number)\ - Probability mass function of the binomial distribution.
- \binomial_probability(n: integer, k: integer, p: integer)\ - Calculate binomial probability P(X=k) = C(n,k)·p^k·(1-p)^(n-k).
- \bootstrap_mean_ci(data: array, confidence: number = 0.95, n_resamples: integer = 1000, seed: integer = None)\ - Estimates a confidence interval for the mean via bootstrap resampling.
- \bowley_skewness(data: array)\ - Bowley (quartile) skewness: (Q3 + Q1 − 2·Q2) / (Q3 − Q1).
- \calculate_covariance(data1: array, data2: array, sample: boolean = True)\ - Calculates the covariance between two lists of numbers.
- \calculate_frecuency(data: array)\ - Calculates the frequency of each value in a list of numbers.
- \calculate_interquartile_range(data: array)\ - Calculates the Interquartile Range (IQR) of a list of numbers.
- \calculate_mean(data: array)\ - Calculates the arithmetic mean (average) of a list of numbers.
- \calculate_median(data: array)\ - Calculates the median (middle value) of a list of numbers.
- \calculate_mode(data: array)\ - Calculates the mode(s) of a list of numbers.
- \calculate_pearson_correlation(data1: array, data2: array)\ - Calculates the Pearson product-moment correlation coefficient between two lists of numbers.
- \calculate_percentile(data: array, percentile: number)\ - Calculates the specified percentile of a list of numbers.
- \calculate_range(data: array)\ - Calculates the range of a list of numbers.
- \calculate_standard_deviation(data: array, sample: boolean = True)\ - Calculates the standard deviation of a list of numbers.
- \calculate_variance(data: array, sample: boolean = True)\ - Calculates the variance of a list of numbers.
- \cauchy_pdf(x: number, x0: number = 0.0, gamma: number = 1.0)\ - Compute the Cauchy distribution PDF.
- \celu(x: number, alpha: number = 1.0)\ - Continuously Differentiable Exponential Linear Unit (CELU).
- \chi_squared_pdf(x: number, k: integer)\ - Compute the chi-squared distribution PDF.
- \chi_squared_statistic(observed: array, expected: array)\ - Chi-squared test statistic: Σ (O − E)² / E.
- \chisq_test(observed: array, expected: array = None)\ - Chi-squared test for independence.
- \coefficient_of_quartile_deviation(data: array)\ - Calculate the Coefficient of Quartile Deviation (CQD).
- \coefficient_of_variation(data: array)\ - Coefficient of variation (CV) — ratio of std dev to mean.
- \cohens_d(group1: array, group2: array)\ - Cohen's d effect size between two independent groups.
- \cohens_kappa_scalar(tp: integer, tn: integer, fp: integer, fn: integer)\ - Calculate Cohen's kappa from a 2×2 confusion matrix.
- \confidence_norm(alpha: number, standard_dev: number, size: integer)\ - Calculates the confidence interval half-width using the normal distribution.
- \confidence_t(alpha: number, standard_dev: number, size: integer)\ - Calculates the confidence interval half-width using the Student's t-distribution.
- \contraharmonic_mean(data: array)\ - Calculate the contraharmonic mean.
- \cosine_similarity(a: array, b: array)\ - Cosine similarity between two vectors.
- \cosine_similarity_scalar(a1: integer, a2: integer, b1: integer, b2: integer)\ - Calculate cosine similarity between two 2D vectors (a1,a2) and (b1,b2).
- \count_true_with_sum(boolean_list: array)\ - Counts the number of True values in a boolean list.
- \covariance_matrix(data: array)\ - Computes the sample covariance matrix for a set of variables.
- \cramers_v(chi2: number, n: integer, k: integer)\ - Cramér's V — effect size for chi-squared tests.
- \cross_entropy(p: array, q: array)\ - Cross-entropy H(P, Q) = −Σ p·log₂(q).
- \cross_entropy_binary(y_true: integer, y_pred: integer)\ - Calculate binary cross-entropy loss for a single observation.
- \cumulative_max(data: array)\ - Returns the running cumulative maximum of a list.
- \cumulative_min(data: array)\ - Returns the running cumulative minimum of a list.
- \cumulative_moving_average(data: array)\ - Cumulative moving average (expanding mean).
- \cumulative_product(data: array)\ - Returns the running cumulative product of a list.
- \cumulative_return(returns: array)\ - Computes the total cumulative return from a list of periodic returns.
- \cumulative_sum(data: array)\ - Returns the running cumulative sum of a list.
- \describe(data: array)\ - Returns a summary of descriptive statistics for a list (like pandas describe).
- \deviation_squared(data: array)\ - Calculates the sum of squared deviations from the mean.
- \dice_coefficient_scalar(tp: integer, fp: integer, fn: integer)\ - Calculate the Sørensen–Dice coefficient from counts: 2·TP / (2·TP + FP + FN).
- \diff(data: array, periods: integer = 1)\ - Calculates consecutive differences in a list (like pandas Series.diff).
- \dixon_q_test(data: array)\ - Perform Dixon's Q test for outlier detection.
- \durbin_watson(residuals: array)\ - Computes the Durbin-Watson statistic for autocorrelation in residuals.
- \elu(x: integer, alpha: number = 1.0)\ - Compute Exponential Linear Unit: x if x > 0, else alpha * (exp(x) - 1).
- \empirical_cdf(data: array, x: number)\ - Computes the empirical cumulative distribution function at x.
- \entropy(data: array)\ - Calculates the Shannon entropy of a discrete dataset.
- \entropy_binary(p: integer)\ - Calculate binary entropy H(p) = -p·log₂(p) - (1-p)·log₂(1-p).
- \erlang_pdf(x: integer, k: integer, lam: integer)\ - Return the Erlang probability density at *x*.
- \euclidean_distance(a: array, b: array)\ - Euclidean distance between two vectors.
- \ewma_variance(data: array, alpha: number = 0.3)\ - Compute exponentially weighted moving variance.
- \exponential_cdf(x: number, lam: number)\ - Compute the exponential distribution CDF.
- \exponential_decay_rate(initial: integer, final: integer, time: integer)\ - Calculate the decay constant λ given initial/final values and time.
- \exponential_moving_average(data: array, alpha: number)\ - Exponential moving average (EMA) with smoothing factor alpha.
- \exponential_pdf(x: number, lam: number)\ - Compute the exponential distribution PDF.
- \exponential_smoothing_single(value: number, previous_smoothed: number, alpha: number)\ - Computes one step of single exponential smoothing.
- \f1_score_scalar(tp: integer, fp: integer, fn: integer)\ - Calculate F1-score from counts: 2·TP / (2·TP + FP + FN).
- \f_test(data1: array, data2: array)\ - F-test for comparing two variances.
- \false_discovery_rate(fp: integer, tp: integer)\ - Calculate False Discovery Rate: FDR = FP / (FP + TP).
- \ffill(data: array)\ - Forward-fills None values with the last non-None value.
- \fillna(data: array, value: any = 0)\ - Replaces None values in a list with a specified fill value.
- \fisher(x: number)\ - Calculates the Fisher transformation.
- \fisher_inv(y: number)\ - Calculates the inverse Fisher transformation.
- \focal_loss(y_true: integer, y_pred: integer, gamma: integer = 2.0, alpha: integer = 0.25)\ - Calculate focal loss for a single sample.
- \forecast_ets(target: integer, values: array, timeline: array, seasonality: integer = 1, data_completion: integer = 1, aggregation: integer = 1)\ - Forecast value using exponential smoothing (ETS).
- \forecast_linear(x: integer, known_y: array, known_x: array)\ - Predicts a value using linear regression.
- \frequency_bins(data: array, bins: array)\ - Calculates frequency distribution of data across bin boundaries.
- \gamma_pdf(x: number, alpha: number, beta: number)\ - Compute the gamma distribution PDF.
- \gelu(x: integer)\ - Compute the Gaussian Error Linear Unit activation.
- \geometric_mean(data: array)\ - Calculates the geometric mean of a list of positive numbers.
- \geometric_pdf(k: integer, p: integer)\ - Return the geometric probability mass at trial *k*.
- \geometric_pmf(k: integer, p: number)\ - Compute the geometric distribution PMF.
- \gini_coefficient(data: array)\ - Computes the Gini coefficient measuring statistical dispersion.
- \gini_impurity_binary(p: integer)\ - Calculate binary Gini impurity G = 2·p·(1-p).
- \glass_delta(treatment: array, control: array)\ - Glass's Δ — effect size using the control group's SD.
- \growth(known_y: array, known_x: array = None, new_x: array = None)\ - Predicted values along an exponential trend.
- \grubbs_test(data: array)\ - Perform Grubbs' test for a single outlier.
- \gumbel_pdf(x: integer, mu: integer = 0.0, beta: integer = 1.0)\ - Return the Gumbel (Type I extreme value) density at *x*.
- \hard_sigmoid(x: number)\ - Hard sigmoid: piecewise-linear approximation of the sigmoid function.
- \hard_swish(x: number)\ - Hard swish: efficient approximation of the swish activation.
- \harmonic_mean(data: array)\ - Calculates the harmonic mean of a list of positive numbers.
- \hedges_g(group1: array, group2: array)\ - Hedges' g — bias-corrected effect size (for small samples).
- \hinge_loss(y_true: integer, y_pred: integer)\ - Compute hinge loss for a single sample.
- \holt_linear_forecast(data: array, alpha: number, beta: number, steps: integer = 1)\ - Holt's linear exponential smoothing forecast.
- \huber_loss(y_true: integer, y_pred: integer, delta: integer = 1.0)\ - Compute the Huber loss for a single prediction.
- \hypergeometric_pmf(k: integer, N: integer, K: integer, n: integer)\ - Compute the hypergeometric distribution PMF.
- \intercept(known_y: array, known_x: array)\ - Calculates the y-intercept of the linear regression line.
- \jaccard_index_scalar(tp: integer, fp: integer, fn: integer)\ - Calculate the Jaccard index from scalar confusion-matrix counts.
- \jaccard_similarity(set_a: array, set_b: array)\ - Computes the Jaccard similarity index between two collections.
- \jarque_bera(data: array)\ - Performs the Jarque-Bera test for normality.
- \js_divergence(p: array, q: array)\ - Jensen-Shannon divergence: symmetric, bounded version of KL.
- \kendall_tau(data1: array, data2: array)\ - Calculates the Kendall rank correlation coefficient (tau-b).
- \kl_divergence(p: array, q: array)\ - Kullback-Leibler divergence D_KL(P || Q) = Σ p·log₂(p/q).
- \kurtosis(data: array, excess: boolean = True)\ - Calculates the kurtosis of a dataset.
- \lag(data: array, periods: integer = 1, default: any = None)\ - Shifts values forward (access previous values), like SQL LAG.
- \laplace_pdf(x: number, mu: number = 0.0, b: number = 1.0)\ - Compute the Laplace distribution PDF.
- \large(data: array, k: integer)\ - Returns the k-th largest value in a dataset.
- \lead(data: array, periods: integer = 1, default: any = None)\ - Shifts values backward (access subsequent values), like SQL LEAD.
- \leaky_relu(x: integer, alpha: number = 0.01)\ - Compute Leaky Rectified Linear Unit: x if x > 0 else alpha * x.
- \linest(known_y: array, known_x: array = None)\ - Linear regression statistics.
- \ljung_box(residuals: array, lags: integer = 10)\ - Compute the Ljung-Box Q statistic for autocorrelation testing.
- \log_cosh_loss(actual: integer, predicted: integer)\ - Calculate the log-cosh loss for a single observation.
- \log_loss(y_true: integer, y_pred: integer, eps: number = 1e-15)\ - Compute binary cross-entropy (log loss) for a single sample.
- \log_normal_pdf(x: number, mu: number = 0.0, sigma: number = 1.0)\ - Compute the log-normal distribution PDF.
- \logest(known_y: array, known_x: array = None)\ - Exponential regression statistics.
- \logistic_cdf(x: number, mu: number = 0.0, s: number = 1.0)\ - Compute the logistic distribution CDF.
- \logistic_function(x: integer, L: integer = 1.0, k: integer = 1.0, x0: integer = 0.0)\ - Calculate the generalised logistic (sigmoid) function.
- \logistic_pdf(x: number, mu: number = 0.0, s: number = 1.0)\ - Compute the logistic distribution PDF.
- \logit(p: integer)\ - Compute the logit (log-odds) function, inverse of sigmoid.
- \mahalanobis_distance_1d(x: number, data: array)\ - Mahalanobis distance of a point from a 1D dataset.
- \manhattan_distance(a: array, b: array)\ - Manhattan (L1) distance between two vectors.
- \mann_whitney_u(data1: array, data2: array)\ - Performs the Mann-Whitney U test (standalone, no scipy).
- \markov_chain_steady_state(transition: array, tol: number = 1e-10, max_iter: integer = 1000)\ - Compute the steady-state distribution of an ergodic Markov chain.
- \matthews_corrcoef_scalar(tp: integer, tn: integer, fp: integer, fn: integer)\ - Calculate Matthews Correlation Coefficient from confusion matrix counts.
- \max_drawdown(values: array)\ - Computes the maximum drawdown of a value series.
- \max_if(values: array, criteria_range: array, criteria: any)\ - Return the maximum of *values* where the parallel *criteria_range* meets *criteria*.
- \maxwell_boltzmann_pdf(v: integer, a: integer)\ - Return the Maxwell-Boltzmann speed distribution density.
- \mean_absolute_error(observed: array, predicted: array)\ - Mean absolute error (MAE).
- \mean_absolute_percentage_error(observed: array, predicted: array)\ - Mean absolute percentage error (MAPE).
- \mean_bias_error(measured: integer, actual: integer)\ - Calculate mean bias error for a single observation: measured - actual.
- \mean_percentage_error(actual: integer, predicted: integer)\ - Calculate mean percentage error for a single observation.
- \mean_squared_error(observed: array, predicted: array)\ - Calculates the mean squared error (MSE) between observed and predicted values.
- \median_absolute_deviation(data: array)\ - Median absolute deviation (MAD) — robust measure of dispersion.
- \midhinge(data: array)\ - Calculate the midhinge of a dataset.
- \midrange(data: array)\ - Calculate the midrange of a dataset.
- \min_if(values: array, criteria_range: array, criteria: any)\ - Return the minimum of *values* where the parallel *criteria_range* meets *criteria*.
- \minkowski_distance(vec_a: array, vec_b: array, p: integer = 2)\ - Computes the Minkowski distance of order p between two vectors.
- \mish(x: integer)\ - Compute Mish activation: x * tanh(softplus(x)).
- \mode_mult(data: array)\ - Returns all modes (most frequent values) in a dataset.
- \mode_single(data: array)\ - Returns the single most frequent value in a dataset.
- \moors_kurtosis(data: array)\ - Moors' octile-based kurtosis measure (robust alternative).
- \moving_median(data: array, window: integer)\ - Computes a rolling median over a sliding window.
- \multinomial_coefficient(counts: array = ...)\ - Compute the multinomial coefficient n! / (k1!·k2!·...·km!).
- \mutual_information(joint: array)\ - Mutual information I(X;Y) from a joint probability table.
- \negative_binomial_pmf(k: integer, r: integer, p: number)\ - Compute the negative binomial distribution PMF.
- \negative_predictive_value(tn: integer, fn: integer)\ - Calculate Negative Predictive Value: NPV = TN / (TN + FN).
- \normal_cdf(x: number, mu: number = 0.0, sigma: number = 1.0)\ - Compute the normal distribution CDF using the error function.
- \normal_pdf(x: number, mu: number = 0.0, sigma: number = 1.0)\ - Probability density function of the normal distribution.
- \odds_ratio(tp: integer, tn: integer, fp: integer, fn: integer)\ - Calculate the odds ratio from a 2×2 confusion matrix.
- \outliers_iqr(data: array, factor: number = 1.5)\ - Detects outliers using the Interquartile Range (IQR) method.
- \pareto_pdf(x: number, x_m: number, alpha: number)\ - Compute the Pareto distribution PDF.
- \partial_autocorrelation(data: array, max_lag: integer)\ - Partial autocorrelation function (PACF) via Durbin-Levinson recursion.
- \pct_change(data: array)\ - Calculates the percentage change between consecutive elements.
- \percentile_exc(data: array, k: number)\ - k-th percentile using exclusive interpolation.
- \percentile_rank(data: array, value: number)\ - Percentile rank of a value within a dataset.
- \percentile_to_z(p: integer)\ - Converts a percentile (0–100) to a z-score using inverse erf approximation.
- \percentile_to_z_score(p: number)\ - Convert a percentile (cumulative probability) to a z-score.
- \percentrank_exc(data: array, x: integer, significance: integer = 3)\ - Rank of a value as a percentage (exclusive).
- \point_biserial_correlation(binary: array, continuous: array)\ - Point-biserial correlation between a binary and continuous variable.
- \poisson_pmf(k: integer, lam: number)\ - Probability mass function of the Poisson distribution.
- \poisson_probability(k: integer, lam: integer)\ - Calculate Poisson probability P(X=k) = e^(-λ) · λ^k / k!.
- \precision_score_scalar(tp: integer, fp: integer)\ - Calculate precision from counts: TP / (TP + FP).
- \probability_range(x_range: array, prob_range: array, lower_limit: number, upper_limit: number = None)\ - Sum of probabilities for values within a range.
- \quantile(data: array, q: number)\ - Calculates the q-th quantile of a dataset using linear interpolation.
- \quartile_deviation(data: array)\ - Quartile deviation (semi-interquartile range): (Q3 − Q1) / 2.
- \quartile_exc(data: array, quart: integer)\ - Quartile using exclusive interpolation.
- \r_squared(observed: array, predicted: array)\ - Coefficient of determination (R²).
- \r_squared_scalar(ss_res: integer, ss_tot: integer)\ - Calculate the coefficient of determination (R²).
- \rank(data: array, method: string = average, ascending: boolean = True)\ - Assigns a rank to each value in a list.
- \rayleigh_pdf(x: integer, sigma: integer = 1.0)\ - Return the Rayleigh probability density at *x*.
- \recall_score_scalar(tp: integer, fn: integer)\ - Calculate recall (sensitivity) from counts: TP / (TP + FN).
- \relative_error(measured: integer, actual: integer)\ - Compute the relative error: |measured - actual| / |actual|.
- \relative_standard_deviation(data: array)\ - Computes the Relative Standard Deviation (coefficient of variation) as a percentage.
- \relu(x: integer)\ - Compute the Rectified Linear Unit: max(0, x).
- \residual_standard_error(observed: array, predicted: array, p: integer = 1)\ - Residual standard error (RSE) of a regression model.
- \risk_ratio(exposed_events: integer, exposed_total: integer, control_events: integer, control_total: integer)\ - Calculate the relative risk (risk ratio): RR = (a/n₁) / (c/n₂).
- \rolling_correlation(data1: array, data2: array, window: integer)\ - Computes rolling Pearson correlation over a sliding window.
- \rolling_mean(data: array, window: integer)\ - Calculates the rolling (moving) average over a fixed window size.
- \rolling_median(data: array, window: integer)\ - Calculates a rolling (moving) median over a dataset.
- \rolling_std(data: array, window: integer, sample: boolean = True)\ - Calculates the rolling (moving) standard deviation over a fixed window.
- \rolling_sum(data: array, window: integer)\ - Calculates the rolling (moving) sum over a fixed window size.
- \root_mean_square(data: array)\ - Calculates the root mean square (RMS) of a list of numbers.
- \root_mean_squared_error(actual: array, predicted: array)\ - Computes the Root Mean Squared Error between actual and predicted values.
- \root_mean_squared_log_error(observed: array, predicted: array)\ - Root mean squared logarithmic error (RMSLE).
- \running_max(data: array, window: integer)\ - Running (rolling) maximum over a sliding window.
- \running_min(data: array, window: integer)\ - Running (rolling) minimum over a sliding window.
- \running_std(data: array, window: integer)\ - Running (rolling) standard deviation over a sliding window (sample).
- \runs_test_statistic(binary_sequence: array)\ - Performs the Wald-Wolfowitz runs test for randomness.
- \shannon_entropy(probabilities: array)\ - Shannon entropy H(X) = −Σ p·log₂(p).
- \shapiro_wilk_w(data: array)\ - Compute the Shapiro-Wilk W statistic for normality testing.
- \sigmoid(x: integer)\ - Compute the logistic sigmoid function σ(x) = 1 / (1 + e⁻ˣ).
- \simple_moving_average(data: array, window: integer)\ - Simple moving average (SMA) over a sliding window.
- \skew_p(data: array)\ - Population skewness of a distribution.
- \skewness(data: array)\ - Calculates the skewness of a dataset.
- \slope(known_y: array, known_x: array)\ - Calculates the slope of the linear regression line.
- \small(data: array, k: integer)\ - Returns the k-th smallest value in a dataset.
- \softplus(x: integer)\ - Compute the softplus activation: softplus(x) = log(1 + eˣ).
- \spearman_correlation(data1: array, data2: array)\ - Calculates the Spearman rank correlation coefficient.
- \spearman_rank_correlation(data1: array, data2: array)\ - Computes Spearman's rank correlation coefficient.
- \specificity_score_scalar(tn: integer, fp: integer)\ - Calculate specificity (true negative rate): TN / (TN + FP).
- \standard_error(data: array)\ - Calculates the standard error of the mean (SEM).
- \standard_error_estimate(known_y: array, known_x: array)\ - Returns the standard error of the predicted y-value for each x.
- \standardize(x: number, mean: number, standard_dev: number)\ - Normalized value from a distribution.
- \sum_if(values: array, criteria_range: array, criteria: any)\ - Return the sum of *values* where the parallel *criteria_range* meets *criteria*.
- \sum_list(data: array)\ - Calculates the sum of all values in a list.
- \sum_of_squared_deviations(data: array)\ - Return the sum of squared deviations from the mean.
- \sum_of_squares(data: array)\ - Calculates the sum of squares of a list of numbers.
- \sum_product(list1: array, list2: array)\ - Return the sum of element-wise products of two lists.
- \swish(x: integer)\ - Compute Swish activation: x * sigmoid(x).
- \symmetric_mape(actual: integer, predicted: integer)\ - Calculate symmetric mean absolute percentage error for a single observation.
- \t_test(data1: array, data2: array, tails: integer = 2, type_: integer = 2)\ - Student's t-test.
- \tanh_activation(x: integer)\ - Compute the hyperbolic tangent activation function.
- \theil_sen_slope(x_data: array, y_data: array)\ - Computes the Theil-Sen slope estimator (median of all pairwise slopes).
- \trend(known_y: array, known_x: array = None, new_x: array = None, const: boolean = True)\ - Predicted values along a linear trend.
- \triangular_pdf(x: number, a: number, b: number, c: number)\ - Compute the triangular distribution PDF.
- \trimmed_mean(data: array, percent: number)\ - Calculates the mean excluding a percentage of outliers from each end.
- \trimmed_variance(data: array, proportion: number = 0.05)\ - Variance of the trimmed dataset (after removing tails).
- \tukey_trimean(data: array)\ - Calculate Tukey's trimean.
- \uniform_cdf(x: number, a: number, b: number)\ - Compute the continuous uniform distribution CDF.
- \uniform_pdf(x: number, a: number, b: number)\ - Compute the continuous uniform distribution PDF.
- \weibull_pdf(x: number, k: number, lam: number)\ - Compute the Weibull distribution PDF.
- \weighted_mean(data: array, weights: array)\ - Calculates the weighted arithmetic mean of a list of numbers.
- \weighted_median(data: array, weights: array)\ - Calculates the weighted median of a dataset.
- \weighted_moving_average(data: array, weights: array)\ - Weighted moving average (WMA) with custom weights.
- \weighted_variance(data: array, weights: array, sample: boolean = True)\ - Calculates the weighted variance of a dataset.
- \winsorize(data: array, percent: number)\ - Replaces extreme values with the corresponding percentile boundaries.
- \winsorized_mean(data: array, proportion: number = 0.05)\ - Winsorized mean: replace extreme values before averaging.
- \z_score(value: number, mean: number, std_dev: number)\ - Standard score (z-score).
- \z_score_single(value: integer, mean: integer, std_dev: integer)\ - Calculate the z-score of a single observation: (x - μ) / σ.
- \z_score_to_percentile(z: number)\ - Approximate percentile from z-score using the cumulative normal.
- \z_test(data: array, x: number, sigma: number = None)\ - One-tailed p-value of a z-test.
- \z_to_percentile(z: integer)\ - Converts a z-score to a percentile (0–100) using the error function.
- \zipf_pmf(k: integer, s: integer, n: integer)\ - Return the Zipf probability mass for rank *k*.

## fxNumeric.tensor_functions

- \christoffel_symbols_diagonal(metric_diag: array, d_metric: array)\ - Christoffel symbols of the second kind for a diagonal metric.
- \geodesic_equation_rhs(gamma: array, velocity: array)\ - Right-hand side of the geodesic equation: d²x^k/dτ² = -Γ^k_{ij} dx^i/dτ dx^j/dτ.
- \kronecker_delta(i: integer, j: integer)\ - Kronecker delta δ_ij: 1 if i == j, else 0.
- \levi_civita(i: integer, j: integer, k: integer)\ - Levi-Civita (permutation) symbol ε_{ijk} for 3D.
- \levi_civita_nd(indices: array)\ - Generalized Levi-Civita symbol for n dimensions.
- \lower_index(v_contravariant: array, metric_diag: array)\ - Lowers the index of a contravariant vector using a diagonal metric.
- \metric_tensor_cylindrical(r: number)\ - Metric tensor for cylindrical coordinates (r, φ, z).
- \metric_tensor_from_jacobian(jacobian: array)\ - Computes the metric tensor g_{ij} = J^T · J from a Jacobian matrix.
- \metric_tensor_spherical(r: number, theta: number)\ - Metric tensor for spherical coordinates (r, θ, φ).
- \raise_index(v_covariant: array, metric_inverse_diag: array)\ - Raises the index of a covariant vector using a diagonal inverse metric.
- \riemann_christoffel_check_2d(gamma: array)\ - Computes the single independent component of the Riemann curvature tensor in 2D.
- \tensor_contract(t: array, idx1: integer = 0, idx2: integer = 1)\ - Contracts (traces) a rank-2 tensor over two indices.
- \tensor_outer_product(u: array, v: array)\ - Outer (tensor) product of two vectors: T_{ij} = u_i · v_j.

## fxNumeric.transform_functions

- \abel_transform_numerical(f: any, y: number, r_max: number = 10.0, n_points: integer = 1000)\ - Numerical Abel transform: F(y) = 2 ∫_y^∞ f(r) r / √(r²-y²) dr.
- \auto_correlation(signal: array)\ - Computes the discrete auto-correlation of a signal.
- \convolution(f: array, g: array)\ - Computes the discrete linear convolution of two sequences.
- \cross_correlation(f: array, g: array)\ - Computes the discrete cross-correlation of two sequences.
- \dft(signal: array)\ - Computes the Discrete Fourier Transform of a real signal.
- \discrete_cosine_transform(signal: array)\ - Computes the Type-II Discrete Cosine Transform (DCT-II).
- \discrete_sine_transform(signal: array)\ - Computes the Type-I Discrete Sine Transform (DST-I).
- \fourier_transform_numerical(f: number, omega: number, t_range: number = (-50.0, 50.0), n_points: integer = 2000)\ - Numerically approximates the Fourier transform F(omega).
- \hankel_transform_numerical(f: any, k: number, nu: number = 0.0, a: number = 1e-06, b: number = 50.0, n: integer = 2000)\ - Computes the Hankel transform H_ν{f}(k) = ∫_0^∞ f(r) J_ν(kr) r dr numerically.
- \hartley_transform(signal: array)\ - Computes the discrete Hartley transform (DHT).
- \hilbert_transform(signal: array)\ - Approximates the discrete Hilbert transform of a real signal.
- \idft(spectrum: array)\ - Computes the Inverse Discrete Fourier Transform.
- \inverse_discrete_cosine_transform(coefficients: array)\ - Computes the inverse Type-II DCT (DCT-III).
- \inverse_hartley_transform(coefficients: array)\ - Computes the inverse discrete Hartley transform.
- \inverse_laplace_gaver_stehfest(f_s: number, t: number, n: integer = 12)\ - Numerically inverts the Laplace transform using the Gaver-Stehfest algorithm.
- \laplace_transform_numerical(f: number, s: number, t_max: number = 50.0, n_points: integer = 1000)\ - Numerically approximates the Laplace transform F(s) = integral_0^inf f(t) e^{-st} dt.
- \mellin_transform_numerical(f: any, s: number, a: number = 1e-06, b: number = 100.0, n: integer = 2000)\ - Computes the Mellin transform M{f}(s) = ∫_0^∞ x^(s-1) f(x) dx numerically.
- \z_transform_eval(x: array, z_real: number, z_imag: number)\ - Evaluates the Z-transform X(z) = sum_{n=0}^{N-1} x[n] z^{-n} at a given z.

## fxNumeric.trigonometry_functions

- \along_track_distance(lat: number, lon: number, path_lat1: number, path_lon1: number, path_lat2: number, path_lon2: number, radius: number = 6371000.0)\ - Distance along the great-circle path from start to the closest point.
- \angle_bisector_length(a: number, b: number, angle: number)\ - Compute the length of the angle bisector in a triangle.
- \angular_deficiency(n: integer)\ - Compute the angular deficiency of a regular polygon vertex.
- \angular_velocity(revolutions: integer, time_seconds: integer)\ - Computes angular velocity in radians per second.
- \annular_sector_area(inner_radius: integer, outer_radius: integer, angle: integer)\ - Return the area of an annular sector.
- \annulus_area(outer_radius: integer, inner_radius: integer)\ - Calculate the area of an annulus (ring): A = π * (R² - r²).
- \arbelos_area(r: integer, r1: integer, r2: integer)\ - Return the area of an arbelos.
- \arc_length(radius: number, angle_radians: number)\ - Compute arc length: r·θ.
- \arccosine(x: number)\ - Calculates the inverse cosine (arccosine) of a value.
- \arcsine(x: number)\ - Calculates the inverse sine (arcsine) of a value.
- \arctangent(x: number)\ - Calculates the inverse tangent (arctangent) of a value.
- \arctangent2(y: number, x: number)\ - Calculates the inverse tangent of y/x using the signs of both arguments to determine the correct quadrant.
- \bearing(lat1: number, lon1: number, lat2: number, lon2: number)\ - Computes the initial bearing (forward azimuth) between two GPS coordinates.
- \cartesian_to_cylindrical(x: number, y: number, z: number)\ - Convert Cartesian coordinates (x, y, z) to cylindrical (ρ, φ, z).
- \cartesian_to_polar(x: number, y: number)\ - Convert Cartesian coordinates (x, y) to polar (r, θ).
- \cartesian_to_spherical(x: number, y: number, z: number)\ - Convert Cartesian coordinates (x, y, z) to spherical (r, θ, φ).
- \chord_length(radius: number, angle_radians: number)\ - Compute the chord length: 2r·sin(θ/2).
- \circular_arc_length(radius: integer, angle: number)\ - Computes the arc length of a circle given radius and central angle.
- \circular_ring_area(outer_radius: integer, inner_radius: integer)\ - Calculate the area of a circular ring (same as annulus): A = π(R² - r²).
- \circular_sector_area(radius: integer, angle: integer)\ - Calculate the area of a circular sector A = ½·r²·θ.
- \circular_segment_area(radius: integer, angle: integer)\ - Calculate the area of a circular segment.
- \circular_segment_chord(radius: integer, angle: integer)\ - Return the chord length of a circular segment.
- \circular_segment_height(radius: number, angle_radians: number)\ - Compute the height of a circular segment (sagitta).
- \cone_lateral_area(radius: integer, slant_height: integer)\ - Calculate the lateral surface area of a cone: A = π·r·l.
- \cone_slant_height(radius: integer, height: integer)\ - Calculate the slant height of a cone l = √(r² + h²).
- \cosecant(angle_radians: number)\ - Calculates the cosecant of an angle.
- \cosine(angle_radians: number)\ - Calculates the cosine of an angle.
- \cotangent(angle_radians: number)\ - Calculates the cotangent of an angle.
- \coversine(angle_radians: number)\ - Computes the coversine: coversin(x) = 1 - sin(x).
- \cross_track_distance(lat: number, lon: number, path_lat1: number, path_lon1: number, path_lat2: number, path_lon2: number, radius: number = 6371000.0)\ - Perpendicular distance from a point to a great-circle path (meters).
- \cyclic_polygon_radius(side_length: integer, n_sides: integer)\ - Return the circumradius of a regular polygon.
- \cylinder_lateral_area(radius: integer, height: integer)\ - Calculate the lateral surface area of a cylinder: A = 2·π·r·h.
- \cylindrical_to_cartesian(rho: number, phi: number, z: number)\ - Convert cylindrical coordinates (ρ, φ, z) to Cartesian (x, y, z).
- \damped_oscillation(amplitude: number, decay: number, frequency: number, t: number)\ - Evaluate a damped oscillation: A·e^(-γt)·cos(2πft).
- \degrees_to_radians(degrees: number)\ - Converts an angle from degrees to radians.
- \destination_point(lat: number, lon: number, brng: number, distance_km: number)\ - Computes the destination point given an origin, bearing, and distance.
- \distance_point_to_line(px: number, py: number, x1: number, y1: number, x2: number, y2: number)\ - Computes the perpendicular distance from a point to a line segment.
- \ellipse_area(semi_major: integer, semi_minor: integer)\ - Calculate the area of an ellipse: A = π * a * b.
- \ellipse_circumference_approx(a: integer, b: integer)\ - Approximate the circumference of an ellipse using Ramanujan's formula.
- \ellipse_perimeter(semi_major: integer, semi_minor: integer)\ - Approximate the perimeter of an ellipse using Ramanujan's formula.
- \ellipsoid_volume(a: integer, b: integer, c: integer)\ - Calculate the volume of an ellipsoid V = (4/3)·π·a·b·c.
- \epicycloid_arc_length(big_r: integer, small_r: integer)\ - Return the total arc length of one full epicycloid.
- \excosecant(angle_radians: number)\ - Computes the excosecant: excsc(x) = csc(x) - 1.
- \exsecant(angle_radians: number)\ - Computes the exsecant: exsec(x) = sec(x) - 1.
- \frustum_lateral_area(r1: integer, r2: integer, slant_height: integer)\ - Return the lateral surface area of a conical frustum.
- \frustum_volume(radius_top: integer, radius_bottom: integer, height: integer)\ - Calculate the volume of a conical frustum.
- \geodesic_area(vertices: array)\ - Approximate area of a polygon on a sphere using the spherical excess formula.
- \gudermannian(x: integer)\ - Compute the Gudermannian function: gd(x) = 2 * atan(tanh(x/2)).
- \hacoverversine(angle_radians: number)\ - Compute the hacoversed versine (hacoverversine) = (1 + sin(θ))/2.
- \haversine_angle(theta: integer)\ - Compute the haversine of an angle: hav(θ) = sin²(θ/2).
- \haversine_distance(lat1: number, lon1: number, lat2: number, lon2: number)\ - Calculates the great-circle distance between two points on Earth.
- \haversine_trig(angle_radians: number)\ - Computes the haversine: hav(x) = (1 - cos(x)) / 2 = sin^2(x/2).
- \herons_formula(a: integer, b: integer, c: integer)\ - Computes the area of a triangle from its three side lengths.
- \hexagon_area(side: integer)\ - Calculate the area of a regular hexagon.
- \hyperbolic_cosecant(x: number)\ - Calculates the hyperbolic cosecant of a value.
- \hyperbolic_cosine(x: number)\ - Calculates the hyperbolic cosine of a value.
- \hyperbolic_cosine_derived(x: number)\ - Calculates the hyperbolic cosine of a value using its exponential definition.
- \hyperbolic_cotangent(x: number)\ - Calculates the hyperbolic cotangent of a value.
- \hyperbolic_distance(x1: number, y1: number, x2: number, y2: number)\ - Compute the hyperbolic distance in the Poincaré half-plane model.
- \hyperbolic_secant(x: number)\ - Calculates the hyperbolic secant of a value.
- \hyperbolic_sine(x: number)\ - Calculates the hyperbolic sine of a value.
- \hyperbolic_sine_derived(x: number)\ - Calculates the hyperbolic sine of a value using its exponential definition.
- \hyperbolic_tangent(x: number)\ - Calculates the hyperbolic tangent of a value.
- \hyperbolic_tangent_derived(x: number)\ - Calculates the hyperbolic tangent of a value using its exponential definition.
- \hypocycloid_arc_length(big_r: integer, small_r: integer)\ - Return the total arc length of one full hypocycloid.
- \hypotenuse(x: number, y: number)\ - Calculates the Euclidean norm, sqrt(x*x + y*y). This is the length of the hypotenuse
- \inscribed_angle(central_angle: integer)\ - Calculate the inscribed angle from a central angle.
- \intermediate_point(lat1: number, lon1: number, lat2: number, lon2: number, fraction: number)\ - Find a point at a given fraction along the great-circle path.
- \inverse_cosecant(x: number)\ - Calculates the inverse cosecant (arccosec) of a value.
- \inverse_cotangent(x: number)\ - Calculates the inverse cotangent (arccotan) of a value.
- \inverse_gudermannian(x: integer)\ - Compute the inverse Gudermannian: gd⁻¹(x) = 2 * atanh(tan(x/2)).
- \inverse_hyperbolic_cosecant(x: number)\ - Calculates the inverse hyperbolic cosecant (arccosech) of a value.
- \inverse_hyperbolic_cosine(x: number)\ - Calculates the inverse hyperbolic cosine (arccosh) of a value.
- \inverse_hyperbolic_cosine_derived(x: number)\ - Calculates the inverse hyperbolic cosine (arccosh) of a value.
- \inverse_hyperbolic_cotangent(x: number)\ - Calculates the inverse hyperbolic cotangent (arccotanh) of a value.
- \inverse_hyperbolic_secant(x: number)\ - Calculates the inverse hyperbolic secant (arcsech) of a value.
- \inverse_hyperbolic_sine(x: number)\ - Calculates the inverse hyperbolic sine (arcsinh) of a value.
- \inverse_hyperbolic_sine_derived(x: number)\ - Calculates the inverse hyperbolic sine (arcsinh) of a value.
- \inverse_hyperbolic_tangent(x: number)\ - Calculates the inverse hyperbolic tangent (arctanh) of a value.
- \inverse_hyperbolic_tangent_derived(x: number)\ - Calculates the inverse hyperbolic tangent (arctanh) of a value.
- \inverse_secant(x: number)\ - Calculates the inverse secant (arcsec) of a value.
- \law_of_cosines_angle(a: integer, b: integer, c: integer)\ - Finds the angle opposite side c using the law of cosines.
- \law_of_cosines_side(a: integer, b: integer, angle_c: number)\ - Finds the third side of a triangle using the law of cosines.
- \law_of_sines(a: integer, angle_a: number, angle_b: number)\ - Finds side b using the law of sines: b = a × sin(B) / sin(A).
- \lens_area(radius: integer, distance: integer)\ - Calculate the area of intersection of two equal circles (lens/vesica).
- \midpoint_geo(lat1: number, lon1: number, lat2: number, lon2: number)\ - Computes the geographic midpoint between two coordinates on a sphere.
- \normalize_angle(angle: number, full_turn: number = 6.283185307179586)\ - Normalize an angle to the range [0, full_turn).
- \paraboloid_volume(radius: integer, height: integer)\ - Calculate the volume of a circular paraboloid.
- \parallelogram_area(base: integer, height: integer)\ - Calculate the area of a parallelogram: A = b · h.
- \point_in_circle(px: number, py: number, cx: number, cy: number, radius: number)\ - Checks if a point (px, py) lies inside a circle centred at (cx, cy).
- \point_in_triangle(px: number, py: number, ax: number, ay: number, bx: number, by: number, cx: number, cy: number)\ - Checks if a point (px, py) is inside the triangle (A, B, C).
- \polar_to_cartesian(r: number, theta: number)\ - Convert polar coordinates (r, θ) to Cartesian (x, y).
- \power_of_point(distance: integer, radius: integer)\ - Return the power of a point with respect to a circle.
- \pyramid_volume(base_area: integer, height: integer)\ - Calculate the volume of a pyramid: V = (1/3)·B·h.
- \radians_to_degrees(radians: number)\ - Converts an angle from radians to degrees.
- \regular_polygon_apothem(side_length: integer, n_sides: integer)\ - Return the apothem of a regular polygon.
- \regular_polygon_area(n_sides: integer, side_length: integer)\ - Calculate the area of a regular polygon: A = (n·s²) / (4·tan(π/n)).
- \regular_polygon_exterior_angle(n_sides: integer)\ - Calculate the exterior angle of a regular polygon.
- \regular_polygon_interior_angle(n_sides: integer)\ - Calculate the interior angle of a regular polygon.
- \regular_polygon_perimeter(n_sides: integer, side_length: integer)\ - Calculate the perimeter of a regular polygon: P = n · s.
- \reuleaux_triangle_area(side: integer)\ - Return the area of a Reuleaux triangle.
- \rhombus_area(diagonal1: integer, diagonal2: integer)\ - Calculate the area of a rhombus: A = (d₁·d₂) / 2.
- \rotation_2d(x: number, y: number, angle: number)\ - Rotates a 2D point (x, y) by an angle in radians around the origin.
- \sagitta(radius: number, angle_radians: number)\ - Compute the sagitta (arrow height): r(1 - cos(θ/2)).
- \salinon_area(r: integer, r1: integer)\ - Return the area of a salinon.
- \secant(angle_radians: number)\ - Calculates the secant of an angle.
- \sector_arc_length(radius: integer, angle: integer)\ - Calculate the arc length of a circular sector: L = r · θ.
- \sector_area(radius: number, angle_radians: number)\ - Compute sector area: r²θ/2.
- \segment_area(radius: number, angle_radians: number)\ - Compute circular segment area: r²(θ - sin(θ))/2.
- \sinc(x: number)\ - Calculates the normalized sinc function: sin(pi*x) / (pi*x).
- \sine(angle_radians: number)\ - Calculates the sine of an angle.
- \sinusoidal_wave(amplitude: number, frequency: number, t: number, phase: number = 0.0)\ - Evaluate a sinusoidal wave: A·sin(2πft + φ).
- \solar_elevation(d: any, latitude: number, longitude: number)\ - Estimate the solar elevation angle for a given datetime and location.
- \spherical_cap_area(sphere_radius: integer, cap_height: integer)\ - Calculate the curved surface area of a spherical cap: A = 2·π·R·h.
- \spherical_cap_volume(sphere_radius: integer, cap_height: integer)\ - Calculate the volume of a spherical cap: V = (π·h²/3)·(3R - h).
- \spherical_law_of_cosines(a: number, b: number, C: number)\ - Compute side c using the spherical law of cosines.
- \spherical_lune_area(radius: integer, angle: integer)\ - Return the surface area of a spherical lune.
- \spherical_sector_volume(radius: integer, height: integer)\ - Calculate the volume of a spherical sector.
- \spherical_to_cartesian(r: number, theta: number, phi: number)\ - Convert spherical coordinates (r, θ, φ) to Cartesian (x, y, z).
- \spherical_wedge_volume(radius: integer, angle: integer)\ - Calculate the volume of a spherical wedge (lune).
- \spherical_zone_area(radius: integer, height: integer)\ - Calculate the surface area of a spherical zone A = 2·π·r·h.
- \spheroid_volume(equatorial_radius: integer, polar_radius: integer)\ - Return the volume of a spheroid (ellipsoid of revolution).
- \stadium_area(radius: integer, straight_length: integer)\ - Return the area of a stadium (discorectangle).
- \stadium_perimeter(radius: integer, straight_length: integer)\ - Return the perimeter of a stadium (discorectangle).
- \tangent(angle_radians: number)\ - Calculates the tangent of an angle.
- \torus_surface_area(major_radius: integer, minor_radius: integer)\ - Calculate the surface area of a torus: A = 4π²·R·r.
- \torus_volume(major_radius: integer, minor_radius: integer)\ - Calculate the volume of a torus: V = 2π²·R·r².
- \trapezoid_area(base1: integer, base2: integer, height: integer)\ - Calculate the area of a trapezoid: A = (b₁ + b₂) · h / 2.
- \triangle_area_sas(a: integer, b: integer, angle_c: number)\ - Computes the area of a triangle given two sides and the included angle.
- \triangle_centroid(ax: number, ay: number, bx: number, by: number, cx: number, cy: number)\ - Computes the centroid (centre of mass) of a triangle.
- \triangle_circumradius(a: number, b: number, c: number)\ - Computes the circumradius of a triangle given its three side lengths.
- \triangle_incircle_radius(a: number, b: number, c: number)\ - Computes the inradius of a triangle given its three side lengths.
- \versine(angle_radians: number)\ - Computes the versine: versin(x) = 1 - cos(x).
- \vincenty_distance(lat1: number, lon1: number, lat2: number, lon2: number)\ - Geodesic distance between two points using Vincenty's formulae.
- \wedge_volume(base_area: integer, height: integer)\ - Calculate the volume of a wedge (triangular prism with half-base).

## fxNumeric.vector_analysis_functions

- \curl_numerical(f_components: array, point: array, h: number = 1e-07)\ - Computes the numerical curl of a 3-D vector field F.
- \directional_derivative(f: any, point: array, direction: array, h: number = 1e-07)\ - Computes the directional derivative of f in the given direction.
- \divergence_numerical(f_components: array, point: array, h: number = 1e-07)\ - Computes the numerical divergence of a vector field F.
- \divergence_theorem_verify(field: number, r_surface: number, u_range: number, v_range: number, x_range: number, y_range: number, z_range: number, n_s: integer = 50, n_v_pts: integer = 20)\ - Verifies the divergence theorem: ∬_S F·dS = ∭_V ∇·F dV.
- \flux_integral_numerical(field: number, r: number, u_range: number, v_range: number, n_u: integer = 50, n_v: integer = 50)\ - Computes a vector flux integral ∬_S F · dS numerically.
- \gradient_numerical(f: any, point: array, h: number = 1e-07)\ - Computes the numerical gradient (nabla f) at a point.
- \hessian_numerical(f: any, point: array, h: number = 1e-05)\ - Computes the numerical Hessian matrix of a scalar function f.
- \jacobian_numerical(f_components: array, point: array, h: number = 1e-07)\ - Computes the numerical Jacobian matrix of a vector-valued function F.
- \laplacian_numerical(f: any, point: array, h: number = 1e-05)\ - Computes the numerical Laplacian of a scalar field f.
- \line_integral_numerical(f_components: array, path_points: array)\ - Numerically approximates a line integral of F along a piecewise-linear path.
- \scalar_triple_product(u: array, v: array, w: array)\ - Scalar triple product u · (v × w) = det([u; v; w]).
- \surface_integral_numerical(f: number, r: number, u_range: number, v_range: number, n_u: integer = 50, n_v: integer = 50)\ - Computes a scalar surface integral ∬_S f(x,y,z) dS numerically.
- \vector_projection(u: array, v: array)\ - Projection of vector u onto vector v: proj_v(u) = (u·v / v·v) v.
- \vector_triple_product(u: array, v: array, w: array)\ - Vector triple product u × (v × w) = v(u·w) - w(u·v).

## fxPython.py_convertions

- \convert_collection(data_collection: array, target_type: array)\ - Converts a list, tuple, or set to a specified target collection type.
- \dict_values_to_list(data_dict: object)\ - Returns the values of a dictionary as a list.
- \dictionary_items_to_list_of_tuples(data_dict: object)\ - Returns the key-value pairs (items) of a dictionary as a list of tuples.
- \dictionary_items_to_set_of_tuples(data_dict: object)\ - Returns the key-value pairs (items) of a dictionary as a set of tuples.
- \dictionary_keys_to_list(data_dict: object)\ - Returns the keys of a dictionary as a list.
- \dictionary_keys_to_set(data_dict: object)\ - Returns the keys of a dictionary as a set.
- \dictionary_to_object(dictionary: object)\ - Converts a dictionary into a simple object.
- \dictionary_to_string(input_dict: object)\ - Converts a dictionary into a string representation.
- \dictionary_values_to_set(data_dict: object)\ - Returns the values of a dictionary as a set, which inherently removes duplicate values.
- \list_of_dicts_to_merged_dict(list_of_dicts: array)\ - Merges a list of dictionaries into a single dictionary.
- \list_of_tuples_to_dict(data_list: array)\ - Converts a list of tuples (where each tuple is a key-value pair) to a dictionary.
- \list_to_set(input_list: array)\ - Converts a given list into a set.
- \list_to_string(lst: array, separator: string =  , use_quotes: boolean = False)\ - Converts a list to its string representation.
- \list_to_tuple(input_list: array)\ - Converts a given list into a tuple.
- \set_to_list(input_set: any)\ - Converts a given set into a list.
- \set_to_string(input_set: any, separator: string =  , use_quotes: boolean = False)\ - Converts a given set to its string representation.
- \set_to_tuple(input_set: any)\ - Converts a given set into a tuple.
- \string_to_dictionary(input_string: string)\ - Converts a string representation back into a dictionary.
- \string_to_list(input_string: string, split_by_character: string = None)\ - Converts a string into a list of substrings or individual characters.
- \tuple_to_list(input_tuple: any)\ - Converts a given tuple into a list.
- \tuple_to_set(input_tuple: any)\ - Converts a given tuple into a set.
- \tuple_to_string(input_tuple: any, separator: string =  , use_quotes: boolean = False)\ - Converts a given tuple to its string representation.

## fxPython.py_logic

- \and_all(args: array = ...)\ - Returns True if all arguments are True; False otherwise.
- \coalesce(values: array = ...)\ - Return the first non-None value from arguments.
- \if_error(value: any, value_if_error: any)\ - Returns value if not an error; otherwise returns alternative value.
- \if_then_else(logical_test: boolean, value_if_true: any, value_if_false: any)\ - Returns one value if condition is True, another if False.
- \ifs(args: array = ...)\ - Evaluates multiple conditions and returns the first matching result.
- \is_blank(value: any)\ - Returns True if value is blank (None, empty string, or empty collection).
- \is_error(value: any)\ - Returns True if evaluating value produces an error.
- \is_logical(value: any)\ - Returns True if value is a boolean.
- \is_number(value: any)\ - Returns True if value is a number (int or float, not bool).
- \is_text(value: any)\ - Returns True if value is a string.
- \not_value(logical_value: boolean)\ - Reverses the logical value of its argument.
- \or_any(args: array = ...)\ - Returns True if any argument is True; False if all are False.
- \switch_case(expression: any, args: array = ...)\ - Evaluates expression against values and returns matching result.
- \xor_all(args: array = ...)\ - Returns True if an odd number of arguments are True.

## fxPython.py_operations

- \add_to_tuple(p_tuple: any, p_value: any)\ - Adds an element to the end of a tuple.
- \calculate(expression_string: any)\ - Evaluates a compound arithmetic expression from a string.
- \choose(index: integer, values: array = ...)\ - Chooses a value from a list of values based on a 1-based index.
- \choose_cols(array: array, col_nums: array = ...)\ - Selects specific columns from a 2-D array.
- \choose_rows(array: array, row_nums: array = ...)\ - Selects specific rows from a 2-D array.
- \chunk(iterable: any, n: integer)\ - Splits an iterable into fixed-size chunks.
- \collection_avg(collection: integer, ignore_none: boolean = True)\ - Calculates the average of the numeric values in an iterable collection.
- \collection_count(collection: any, ignore_none: boolean = True)\ - Counts the number of elements in an iterable collection.
- \collection_filter(collection: any, filter_logic_func: any)\ - Filters an iterable (list, tuple, set, etc.) based on a provided filtering function.
- \collection_max(collection: any, ignore_none: boolean = True)\ - Returns the highest value in an iterable collection.
- \collection_min(collection: any, ignore_none: boolean = True)\ - Returns the smallest value in an iterable collection.
- \collection_stdev(collection: integer, is_sample: boolean = True, ignore_none: boolean = True)\ - Calculates the standard deviation of the values in an iterable collection.
- \collection_sum(collection: integer, ignore_none: boolean = True)\ - Calculates the sum of the numeric values in an iterable collection.
- \combine_dictionaries(dict_one: object, dict_two: object, operation: string)\ - Performs union or intersection operations on two dictionaries.
- \conditional_average(values: array, criteria: integer)\ - Averages values that match a criteria function.
- \conditional_count(values: array, criteria: any)\ - Counts values that match a criteria function.
- \conditional_max(values: array, criteria: integer)\ - Returns the maximum of values that match a criteria function.
- \conditional_min(values: array, criteria: integer)\ - Returns the minimum of values that match a criteria function.
- \conditional_sum(values: array, criteria: integer)\ - Sums values that match a criteria function.
- \count_blank(values: array)\ - Count blank items (``None`` or ``""``).
- \count_by(iterable: any, key_func: any)\ - Count the number of elements in each group defined by a key function.
- \count_if(values: array, criteria: any)\ - Count items that meet an Excel-style criteria.
- \count_ifs(values: array, criteria_pairs: array = ...)\ - Count items where every criteria pair is satisfied.
- \count_numbers(values: array = ...)\ - Count how many items are numeric (int or float, excluding bool).
- \count_values(values: array = ...)\ - Count non-empty, non-None items.
- \deep_flatten(nested: any)\ - Recursively flatten a nested structure of arbitrary depth.
- \deep_merge(dict1: object, dict2: object)\ - Recursively merges two dictionaries.
- \dictionary_filter_by_keys(input_dict: object, keys_to_include: array)\ - Filters a dictionary, returning a new dictionary with only the specified keys.
- \dictionary_rename_keys(original_dict: object, key_mapping: object)\ - Renames keys in a dictionary based on a provided mapping.
- \drop_from_array(array: array, rows: integer = 0, columns: integer = 0)\ - Drops rows and/or columns from a 2-D array.
- \expand_array(array: array, rows: integer = None, columns: integer = None, pad_with: any = None)\ - Expand a 2-D array to target dimensions, padding new cells.
- \filter_elements_by_another(main_collection: any, filter_criteria: any, return_type: string = list)\ - Filters elements from a main collection that are also present in a filter criteria collection.
- \find(predicate: any, iterable: any, default: any = None)\ - Returns the first element matching a predicate.
- \flatten_dict(d: object, separator: string = ., parent_key: string = )\ - Flattens a nested dictionary into a single-level dictionary.
- \flatten_list(p_list: array)\ - Flattens a list of lists into a single, one-dimensional list.
- \frequencies(items: any)\ - Counts occurrences of each element in an iterable.
- \get_nested(data: object, keys: array, default: any = None)\ - Retrieves a value from a nested dictionary using a sequence of keys.
- \group_by(iterable: any, key_func: any)\ - Groups elements of an iterable by a key function.
- \hlookup(lookup_value: any, table: array, row_index: integer, approximate: boolean = False)\ - Searches for a value in the first row and returns a value from another row.
- \hstack(arrays: array = ...)\ - Horizontally concatenates 2-D arrays side by side.
- \index_2d(array: array, row_num: integer, col_num: integer = None)\ - Returns an element or row/column from a 2-D array.
- \index_by(dicts: array, key: string)\ - Index a list of dicts by a given field, creating a lookup table.
- \is_subsequence(sub_sequence: any, main_sequence: any)\ - Checks if 'sub_sequence' is a subsequence of 'main_sequence', maintaining the order of elements.
- \list_has_list(container_list: array, search_list: array)\ - Checks if all elements from one list exist in another list, regardless of order.
- \list_intersection(p_list1: array, p_list2: array)\ - Finds the intersection of two lists, returning a set of common elements.
- \make_array(rows: integer, columns: integer, fn: integer)\ - Generates a 2-D array using a function.
- \merge_elements(first_collection: any, second_collection: any, return_type: string = list, remove_duplicates: boolean = False)\ - Merges two collections into a single collection of the specified type.
- \merge_json_strings(json_str_1: string, json_str_2: string)\ - Merges two JSON strings (representing JSON objects) into a new JSON string.
- \partition(predicate: any, iterable: any)\ - Splits an iterable into two lists based on a predicate.
- \pick_in_collection(collection: any, prompt: string = Por favor, selecciona una opción:, allow_multiple: boolean = False, return_index: boolean = False, exit_on_cancel: boolean = False)\ - Permite al usuario seleccionar una o varias opciones de cualquier colección iterable numerada.
- \pluck(iterable: any, key: any)\ - Extracts a single field from each element in a collection.
- \search(collection: any, target_element: any)\ - Searches for a target element within an iterable collection.
- \sequence(rows: integer = 1, columns: integer = 1, start: number = 1, step: number = 1)\ - Generates a 2-D array of sequential numbers.
- \sort_by(data: array, sort_keys: array, reverse: boolean = False)\ - Sorts data based on corresponding sort keys.
- \sort_dict_by_key(d: object, reverse: boolean = False)\ - Return a new dict sorted by keys.
- \sort_dict_by_value(d: object, reverse: boolean = False)\ - Return a new dict sorted by values.
- \sort_dicts_by_key(items: array, key: string, reverse: boolean = False)\ - Sorts a list of dictionaries by a specific key.
- \take_from_array(array: array, rows: integer = None, columns: integer = None)\ - Takes the first or last rows/columns from a 2-D array.
- \tocol(array: array, scan_by_column: boolean = False)\ - Flatten a 2-D array into a single-column list of lists.
- \torow(array: array, scan_by_column: boolean = False)\ - Flatten a 2-D array into a single-row list.
- \unflatten_dict(d: object, separator: string = .)\ - Reconstructs a nested dictionary from a flat one.
- \unique_list(p_list: array)\ - Returns a new list containing only the unique elements from the original list.
- \unique_tuple_list(p_list: array)\ - Removes duplicate tuples from a list based on the first element of each tuple.
- \vlookup(lookup_value: any, table: array, col_index: integer, approximate: boolean = False)\ - Searches for a value in the first column and returns a value from another column.
- \vstack(arrays: array = ...)\ - Vertically stacks 2-D arrays on top of each other.
- \wrap_rows(vector: array, wrap_count: integer, pad_with: any = None)\ - Wraps a 1-D vector into a 2-D array by rows.
- \xlookup(lookup_value: any, lookup_array: array, return_array: array, if_not_found: any = None, match_mode: integer = 0)\ - Searches a lookup array and returns the corresponding value from a return array.
- \xmatch(lookup_value: any, lookup_array: array, match_mode: integer = 0, search_mode: integer = 1)\ - Returns the 1-based position of a value in an array.
- \zip_dict(keys: array, values: array)\ - Create a dictionary from two parallel lists of keys and values.

## fxPython.py_tools

- \apply_expression(expression: string, iterable: any)\ - Applies a string expression to every item in an iterable and returns a list. Attribute access restricted to built-in types; private attributes blocked.
- \create_key_value_dictionary(p_key_columns: any, p_values: any)\ - Creates a dictionary by mapping key column names to their corresponding values.
- \function_call(func: any, args: array = ..., kwargs: any)\ - Higher-order function that executes a callable with optional arguments.
- \loop_for(iterable: any, func: any)\ - Executes a function for each item in an iterable, simulating a for loop.
- \loop_while(condition: any, func: any, args: array = ..., kwargs: any)\ - Executes a function repeatedly as long as a specified condition is true.
- \pipe(value: any, functions: array = ...)\ - Threads a value through a sequence of functions.
- \retry(func: any, max_attempts: integer = 3, delay: number = 1.0)\ - Retries a function on failure up to a maximum number of attempts.
- \rotate(iterable: any, steps: any = 1)\ - Rotates the elements of an iterable.
- \shift(iterable: any, direction: string = None, new_elements: any = None)\ - Shifts an iterable, adding or removing elements.
- \switch_case(value: any, cases: array = ..., default: any = None)\ - Acts like a generic switch-case statement.

## fxString.string_caseconv

- \to_camel_case(text: string)\ - Converts a string to camelCase.
- \to_constant_case(text: string)\ - Converts a string to CONSTANT_CASE (screaming snake case).
- \to_kebab_case(text: string)\ - Converts a string to kebab-case.
- \to_pascal_case(text: string)\ - Converts a string to PascalCase.
- \to_slug(text: string, separator: string = -)\ - Converts a string to a URL-friendly slug.
- \to_snake_case(text: string)\ - Converts a string to snake_case.
- \to_title_case(text: string)\ - Converts a string to smart Title Case.

## fxString.string_compression

- \compress_string(text: string, encoding: string = utf-8)\ - Compresses a string using zlib and returns a Base64-encoded result.
- \decompress_string(compressed_text: string, encoding: string = utf-8)\ - Decompresses a Base64-encoded zlib-compressed string.

## fxString.string_convertions

- \base64_decode(encoded: string, encoding: string = utf-8)\ - Decode a Base64 string to plain text.
- \base64_encode(text: string, encoding: string = utf-8)\ - Encode a string to Base64.
- \binary_to_text(binary_str: string, separator: string =  )\ - Converts a binary string back to text.
- \boolean_to_string(value: boolean, language: string = en)\ - Converts a boolean value to its string representation.
- \char_from_code(code_point: integer)\ - Returns the Unicode character for a given code point.
- \code_from_char(character: string)\ - Returns the Unicode code point of the first character of a string.
- \extract_and_decode_json(text_content: string)\ - Extracts and decodes a JSON string embedded within a block delimited
- \hex_color_to_rgb(hex_color: string)\ - Convert a hex colour string to an RGB tuple.
- \hex_to_text(hex_str: string, encoding: string = utf-8)\ - Converts a hexadecimal string back to text.
- \integer_to_roman(number: integer)\ - Converts a positive integer to its Roman numeral representation.
- \morse_to_text(morse: string)\ - Decodes International Morse Code back to text.
- \nato_phonetic_to_text(phonetic: string)\ - Convert NATO phonetic alphabet words back to plain text.
- \none_to_string(value: string)\ - Converts a None value to an empty string; otherwise, returns the original value.
- \ordinal_suffix(n: integer)\ - Converts an integer to its English ordinal string.
- \parse_text_to_number(text: string, decimal_separator: string = ., group_separator: string = ,)\ - Convert formatted text to a numeric value.
- \replace_void(primary_value: any, replacement_value: any = NaN)\ - Replaces empty or None values with a specified replacement value.
- \rgb_to_hex_color(r: integer, g: integer, b: integer)\ - Convert RGB values to a hex colour string.
- \roman_to_integer(text: string)\ - Converts a Roman numeral string to an integer.
- \string_to_boolean(text: string)\ - Converts a string representation to a boolean value.
- \string_to_date(input_value: string)\ - Converts a string to a date or datetime object based on common formats.
- \string_to_datetime(input_value: string)\ - Description:
- \string_to_float(input_string: string)\ - Converts a string to a float, returning None if conversion fails.
- \string_to_integer(input_string: string)\ - Converts a string to an integer, returning None if conversion fails.
- \string_to_list(text: string, delimiter: string = ,, strip: boolean = True)\ - Parses a delimited string into a list of strings.
- \string_to_number(input_string: string, target_type: string = string)\ - Converts a string to an integer or float based on the specified target type.
- \text_to_binary(text: string, separator: string =  )\ - Converts text to its binary (8-bit) representation.
- \text_to_braille(text: string)\ - Convert ASCII text to Unicode Braille representation.
- \text_to_hex(text: string, encoding: string = utf-8)\ - Converts text to its hexadecimal byte representation.
- \text_to_morse(text: string)\ - Converts text to International Morse Code.
- \text_to_nato_phonetic(text: string)\ - Convert text to NATO phonetic alphabet representation.
- \text_to_phonetic_ipa(text: string)\ - Convert ASCII text to a simplified IPA transcription.
- \to_full_width(text: string)\ - Converts half-width (single-byte) characters to full-width (double-byte).
- \to_half_width(text: string)\ - Converts full-width (double-byte) characters to half-width (single-byte).
- \value_to_text(value: any, format_type: integer = 0)\ - Converts any value to its text representation.

## fxString.string_encoding

- \caesar_cipher(text: string, shift: integer, decrypt: boolean = False)\ - Applies the Caesar (shift) cipher to a string.
- \decode_base64(encoded_text: string, encoding: string = utf-8)\ - Decodes a Base64-encoded string back to plain text.
- \decode_html_entities(encoded_text: string)\ - Unescapes HTML entities back to their original characters.
- \decode_url(encoded_text: string)\ - Decodes a URL percent-encoded string back to plain text.
- \encode_base64(text: string, encoding: string = utf-8)\ - Encodes a string to Base64 representation.
- \encode_html_entities(text: string)\ - Escapes HTML special characters into their entity equivalents.
- \encode_url(text: string, safe: string = )\ - Encodes a string using URL percent-encoding (RFC 3986).
- \vigenere_cipher(text: string, key: string, decrypt: boolean = False)\ - Applies the Vigenère polyalphabetic cipher to a string.

## fxString.string_evaluations

- \automated_readability_index(text: string)\ - Compute the Automated Readability Index (ARI) for English text.
- \avg_word_length(text: string)\ - Compute the average word length in *text*.
- \char_frequency(text: string)\ - Calculates the frequency of each character in a string.
- \check_password_strength(password: string)\ - Evaluate password strength on a 0–100 score scale.
- \check_substring_at_position(main_string: string, substring: string, position: integer)\ - Checks if a substring exists at a specified position within a main string.
- \coleman_liau_index(text: string)\ - Compute the Coleman–Liau readability index for English text.
- \compare_strings(string1: string, string2: string, case_sensitive: boolean = True)\ - Compare two strings lexicographically.
- \count_sentences(text: string)\ - Count the number of sentences in a text.
- \detect_quotes(text: string)\ - Detects if a string is enclosed in either single or double quotes.
- \domain_from_email(email_address: string)\ - Extracts the domain part from a given email address string.
- \email_belongs_to_name(name: any, email: any)\ - 
- \ends_with_substring(input_string: string, suffix: string, case_sensitive: boolean = True)\ - Checks if a string ends with a specific substring.
- \flesch_reading_ease(text: string)\ - Estimate the Flesch Reading Ease score for English text.
- \get_phones(phone: any, p_separator: any = )\ - Extracts potential phone numbers from a string.
- \get_postalcode(address: string)\ - Extracts 5-digit or 4-digit postal codes from an address string.
- \gunning_fog_index(text: string)\ - Estimate the Gunning Fog readability index for English text.
- \has_date_format(value: string)\ - Verifies if a given string matches any of a predefined list of common date and datetime formats.
- \has_numbers(input_string: string)\ - Checks if the input string contains any numeric digits (0-9).
- \has_substring(input_string: string, char_to_find: string)\ - Checks if a specific character is present within a given string.
- \is_aZ(aZ_string: any)\ - 
- \is_aZ09(aZ0_string: any)\ - 
- \is_alphabetic(alpha_string: any)\ - Checks if a given input is a single alphabetic character (letter),
- \is_anagram(a: string, b: string)\ - Checks whether two strings are anagrams of each other.
- \is_balanced_brackets(text: string)\ - Check whether all brackets in *text* are properly balanced and nested.
- \is_credit_card_format(text: string)\ - Validate a credit card number using the Luhn algorithm.
- \is_email_format(email_string: string)\ - Checks if a given string has the general format of an email address.
- \is_heterogram(text: string)\ - Check if *text* has no repeated characters at all (including spaces).
- \is_internet_domain_format(domain_string: string)\ - Checks if a given string adheres to the general format of an internet domain name.
- \is_ipv4(text: string)\ - Check whether a string is a valid IPv4 address.
- \is_ipv6(text: string)\ - Check whether a string is a valid IPv6 address.
- \is_isogram(text: string)\ - Check if *text* is an isogram (no repeating letters, case-insensitive).
- \is_json(text: string)\ - Check whether a string is valid JSON.
- \is_lipogram(text: string, letter: string)\ - Check if *text* avoids *letter* entirely (case-insensitive).
- \is_numeric(input_string: string)\ - Checks if a given string represents a numeric value (integer or float).
- \is_palindrome(text: string)\ - Checks whether a text is a palindrome (ignoring case, spaces and punctuation).
- \is_pangram(text: string, alphabet: string = abcdefghijklmnopqrstuvwxyz)\ - Checks whether a text contains every letter of the given alphabet.
- \is_tautogram(text: string)\ - Check if every word in *text* starts with the same letter (case-insensitive).
- \is_url_format(url_string: string)\ - Checks if a given string has the general format of a URL.
- \is_uuid(text: string)\ - Check whether a string matches the UUID format (any version).
- \is_valid_base64(text: string)\ - Check whether a string is valid Base64 encoding.
- \is_valid_cron(expr: string)\ - Validate a 5-field cron expression.
- \is_valid_cusip(text: string)\ - Validate a CUSIP (Committee on Uniform Securities Identification Procedures) code.
- \is_valid_domain_format(domain_string: string)\ - Checks if a string conforms to a common internet domain name format.
- \is_valid_e164_phone(text: string)\ - Validate an E.164 international phone number.
- \is_valid_ean(text: string)\ - Validates an EAN-8 or EAN-13 barcode number.
- \is_valid_email_format(email_string: string)\ - Checks if a string has a valid email format.
- \is_valid_hex_color(text: string)\ - Check whether *text* is a valid hex colour code.
- \is_valid_iban(text: string)\ - Validates an International Bank Account Number (IBAN).
- \is_valid_ipv4(text: string)\ - Validate an IPv4 address string.
- \is_valid_ipv6(text: string)\ - Validate an IPv6 address string.
- \is_valid_isbn(text: string)\ - Validates an ISBN-10 or ISBN-13 string.
- \is_valid_isin(text: string)\ - Validate an ISIN (International Securities Identification Number).
- \is_valid_issn(text: string)\ - Validate an ISSN (International Standard Serial Number).
- \is_valid_luhn(text: string)\ - Validates a numeric string using the Luhn (mod-10) algorithm.
- \is_valid_mac_address(text: string)\ - Validate a MAC address string.
- \is_valid_mime_type(text: string)\ - Validates whether a string matches the MIME type format.
- \is_valid_regex(pattern: string)\ - Checks whether a string is a syntactically valid regular expression.
- \is_valid_sedol(text: string)\ - Validate a SEDOL (Stock Exchange Daily Official List) code.
- \is_valid_semver(text: string)\ - Validate a Semantic Versioning string (SemVer 2.0.0).
- \is_valid_swift_bic(text: string)\ - Validates a SWIFT/BIC (Business Identifier Code) string.
- \is_valid_uuid(text: string)\ - Validate a UUID string (versions 1-5).
- \is_valid_vin(text: string)\ - Validate a VIN (Vehicle Identification Number).
- \parse_email(email_address: string)\ - Parses an email address string and returns its username and domain.
- \reading_time(text: string, words_per_minute: integer = 200)\ - Estimates the reading time of a text in minutes.
- \sentence_count(text: string)\ - Counts the number of sentences in a text string.
- \smog_index(text: string)\ - Compute the SMOG readability index for English text.
- \starts_with_substring(input_string: string, prefix: string, case_sensitive: boolean = True)\ - Checks if a string starts with a specific substring.
- \text_entropy(text: string)\ - Calculates Shannon entropy of a string in bits per character.
- \text_stats(text: string)\ - Compute summary statistics for a text string.
- \username_from_email(email_address: string)\ - Extracts and returns a list of individual parts from an email's username.
- \validate_nif_format_and_type(raw_nif_string: string, assume_spanish_if_no_prefix: boolean = True)\ - Validates the format of a NIF (Spanish DNI/NIE/CIF or other European NIFs/VAT IDs).
- \validate_substring_type(original_string: string, extract_method: string, expected_type: string, start_position: integer = None, length: integer = None, num_chars: integer = None, date_format: string = None)\ - Extrae una subcadena y verifica si su contenido corresponde a un tipo de dato esperado.
- \word_frequency(text: string)\ - Calculates the frequency of each word in a text string.

## fxString.string_format

- \apply_string_format(input_string: string, format_code: integer)\ - Applies a specified casing format to a given string.
- \ascii_string(input_string: string)\ - Converts a string to its ASCII representation, ignoring characters that cannot be encoded in ASCII.
- \auto_format_string(input_string: string)\ - Automatically formats a string to either its original case (if all upper/lower)
- \autoformat(data_input: any, locale: string = en_US)\ - Automatically formats an input value as a date or a number based on its type
- \camel_to_snake(text: string)\ - Converts a camelCase or PascalCase string to snake_case.
- \capitalize_string(input_string: string, mode: string = all)\ - Capitalizes words in a string, with an optional mode to handle specific particles.
- \dollar(number: integer, decimals: integer = 2)\ - Format number as currency text with dollar sign.
- \fixed(number: integer, decimals: integer = 2, no_commas: boolean = False)\ - Format number with a fixed number of decimal places.
- \flat_vowels(input_string: string)\ - Removes only acute accents from vowels while preserving 'ñ', 'ç', and 'ü'.
- \format_as_currency(number: number, decimals: integer = 2, symbol: string = $)\ - Formats a number as currency text with thousands separator.
- \format_as_number(number: integer, decimals: integer = 2, use_parens_for_negative: boolean = True)\ - Format a number with thousands separators and optional parentheses for negatives.
- \format_as_percent(number: integer, decimals: integer = 2, use_parens_for_negative: boolean = True)\ - Format a number as a percentage string (multiplied by 100).
- \format_company_name(company_name: string, company_type: string, format_style: string)\ - Formats a company name by appending its type according to a specified style.
- \format_date(date_input: string, output_format: string = None, locale: string = en_US)\ - Formats a date input (string or datetime object) into a specified output format,
- \format_email_address(email_string: string)\ - Cleans an email address string by removing common typos and invalid characters.
- \format_file_size(size_bytes: integer, binary: boolean = True)\ - Convert a byte count to a human-readable file-size string.
- \format_fullname(fullname: string, uppercase: boolean = True)\ - Formats a person's full name by removing unwanted characters and normalizing spaces.
- \format_internet_domain(domain_string: string)\ - Cleans and normalizes an internet domain string.
- \format_list_as_sentence(items: array, conjunction: string = and, separator: string = , )\ - Formats a list of strings as a human-readable sentence.
- \format_name(input_string: string, add_charset: string = , name_type: string = PERSONA, shift: boolean = False)\ - Description:
- \format_number(number_input: integer, decimal_places: integer = 2, locale: string = en_US, currency_symbol: string = )\ - Formats a numeric input (int, float, or string) according to specified locale conventions.
- \format_number_with_negative_style(number_input: integer, decimal_places: integer = 2, locale: string = en_US, currency_symbol: string = , negative_style: string = minus)\ - Formats a numeric input, handling negative numbers with either a minus sign or parentheses.
- \format_url(url_string: string)\ - Cleans a URL string by removing common typos and unsafe/invalid characters.
- \format_with_pattern(value: any, pattern: string)\ - Format a numeric or date value using an Excel/VBA-style pattern.
- \get_string_format(input_string: string)\ - Determines the casing format of a given string.
- \mask_data(input_string: string, mask_char: string = *, num_chars: integer = None, position: string = all, start_index: integer = 0, keep_visible: integer = 0)\ - Mask sensitive data based on various patterns.
- \normalize_spaces(text: string)\ - Normalizes whitespace in a string: replaces multiple spaces with a single space
- \normalize_symbols(text: string)\ - Normalizes spacing around common symbols and punctuation marks in a string.
- \normalize_text(text: string)\ - Normalizes text by removing accents and converting to lowercase.
- \numbers_from_string(input_string: string, separator: string = )\ - Extracts all numerical sequences from a given string and returns them as a single string.
- \pad_string(text: string, length: integer, char: string =  , direction: string = right)\ - Rellena una cadena de texto hasta una longitud específica con un carácter dado.
- \parse_company(company_name: string, legal_forms_set_override: any = None)\ - Description:
- \pluralize_count(count: integer, singular: string, plural: string = None)\ - Returns a count with its noun in singular or plural form.
- \remove_numbers_from_string(input_string: string)\ - Removes all numerical digits and decimal points from a given string.
- \reorder_comma_fullname(name_with_surname: string)\ - Reformats a name string from 'Surname, Name' to 'Name Surname'.
- \rot13(text: string)\ - Applies the ROT13 substitution cipher to a string.
- \snake_to_camel(text: string, pascal: boolean = False)\ - Converts a snake_case string to camelCase (or PascalCase).
- \sql_quote(sql_string: string, db_type: string)\ - Escapes single quotes in a SQL string based on the database type.
- \string_aZ(input_string: any)\ - Feature Description:
- \string_aZ09(input_string: any)\ - Feature Description:
- \string_aZ09_plus(input_string: any, additional_charset: any = )\ - Filters a string to keep only alphanumeric characters (a-z, A-Z, 0-9)
- \string_aZ_plus(input_string: any, additional_charset: any = )\ - Filters a string to keep only alphabetic characters (a-z, A-Z)
- \swap_case(text: string)\ - Inverts the case of every character in a string.
- \to_lower(text: string)\ - Convierte una cadena de texto a minúsculas.
- \to_upper(text: string)\ - Convierte una cadena de texto a mayúsculas.
- \word_wrap(text: string, width: integer = 80)\ - Wraps text to a specified line width, breaking at word boundaries.
- \zfill(text: string, width: integer)\ - Pads a string with leading zeros to fill a given width.

## fxString.string_hashing

- \fingerprint(text: string)\ - Generates a normalized fingerprint for deduplication.
- \hash_string(text: string, algorithm: string = sha256)\ - Generates a hexadecimal hash digest of a string.

## fxString.string_operations

- \abbreviate(text: string, separator: string = )\ - Generates an abbreviation (initials/acronym) from a text string.
- \add_quotes(text: string, quote_type: string = ')\ - Adds single or double quotes to the beginning and end of a string.
- \ascii_from_char(character: string)\ - Devuelve el valor entero (código Unicode/ASCII) del primer carácter de una cadena.
- \caesar_cipher(text: string, shift: integer)\ - Apply Caesar cipher with the given shift (positive = right).
- \camel_to_snake(text: string)\ - Convert camelCase or PascalCase to snake_case.
- \center_string(text: string, width: integer, fill_char: string =  )\ - Centers a string within a given width using a fill character.
- \char_from_ascii(integer_code: integer)\ - Devuelve el carácter correspondiente a un valor entero (código Unicode/ASCII).
- \chunk_string(text: string, size: integer)\ - Split *text* into chunks of *size* characters.
- \clean_non_printable(text: string)\ - Removes all non-printable characters (ASCII 0-31) from a string.
- \clean_nonprintable(text: string)\ - Remove all nonprintable characters (ASCII 0–31) from text.
- \collapse_whitespace(text: string)\ - Replace consecutive whitespace with a single space and strip edges.
- \cologne_phonetic(text: string)\ - Compute the Kölner Phonetik (Cologne Phonetic) code.
- \common_substring(string1: string, string2: string, min_longest_substring: integer = 4)\ - Finds the longest common substring between two input strings, ignoring case,
- \concatenate_strings(string1: string, string2: string)\ - Concatenates two strings into a single string.
- \constant_case(text: string)\ - Convert whitespace-separated *text* to CONSTANT_CASE.
- \count_characters(input_string: string, target_char: string)\ - Counts the occurrences of a specific character within a string.
- \count_consonants(text: string, lang: string = en)\ - Counts the number of consonants in a text string.
- \count_lines(text: string)\ - Counts the number of lines in a text string.
- \count_occurrences(text: string, sub: string)\ - Count non-overlapping occurrences of *sub* in *text*.
- \count_syllables(text: string, lang: string = en)\ - Estimates the total number of syllables in a text.
- \count_vowels(text: string, lang: string = en)\ - Count vowels in a string.
- \count_words(input_string: string)\ - Counts the number of words in a given string.
- \dedent_text(text: string)\ - Remove common leading whitespace from all lines.
- \deduplicate_words(text: string)\ - Removes duplicate words from a text, preserving first occurrence order.
- \distinct_split(text: string, separator: string = ;, strip_tokens: boolean = True, case_sensitive: boolean = True)\ - Split a delimited string, remove duplicate tokens, and re-join.
- \distinct_words(input_string: string, case_sensitive: boolean = False)\ - Extracts all unique words from a string, with an option for case sensitivity.
- \double_metaphone(text: string)\ - Compute the Double Metaphone phonetic encoding.
- \erase_allspaces(text: string)\ - Removes all whitespace characters from the input string.
- \erase_between_delimiters(text: string, delimiter_key: string)\ - Removes text content that is enclosed between a specified pair of delimiters.
- \erase_digits(text: string)\ - Removes all digit characters from the input string.
- \erase_lrspaces(text: string)\ - Removes leading and trailing whitespace from the input string.
- \erase_lspaces(text: string)\ - Removes leading (left) whitespace from the input string.
- \erase_rspaces(text: string)\ - Removes trailing (right) whitespace from the input string.
- \erase_specialchar(text: string, allow_spaces: boolean = True, allow_underscores: boolean = False, additional_allowed_chars: string = )\ - Limpia una cadena de texto eliminando caracteres especiales.
- \erase_symbol(p_iparse: string)\ - Removes a predefined set of common symbols from the input string,
- \extract_content_by_encloser_type(input_string: string, encloser_type: string)\ - Extracts the substring enclosed within a specific type of encloser
- \extract_domain_from_url(url: string)\ - Extracts the domain (hostname) from a URL string.
- \extract_emails(text: string)\ - Extracts all email addresses from a text string.
- \extract_first_number(input_string: string)\ - Extracts the first integer or float (positive or negative) from a string.
- \extract_hashtags(text: string)\ - Extracts ``#hashtag`` tokens from a text string.
- \extract_last_number(input_string: string)\ - Extracts the last integer or float (positive or negative) from a string.
- \extract_mentions(text: string)\ - Extracts ``@mention`` tokens from a text string.
- \extract_numbers(input_string: string)\ - Extracts all integers and floats (positive or negative) from a string.
- \extract_urls(text: string)\ - Extracts all URLs from a text string.
- \generate_initials(name: string, separator: string = .)\ - Generates initials from a person's name.
- \generate_password(length: integer = 16, uppercase: boolean = True, lowercase: boolean = True, digits: boolean = True, special: boolean = True)\ - Generate a cryptographically secure random password.
- \get_in_text_by_pattern(text: string, pattern_type: string)\ - Finds and returns all occurrences of a specified pattern type within the given text.
- \get_line(text: string, line_number: integer)\ - Returns a specific line from a multiline string (1-indexed).
- \get_substrings(string1: string, string2: string)\ - Identifies and returns all common contiguous substrings (matching blocks)
- \indent_text(text: string, prefix: string =   )\ - Add a prefix string to the beginning of every line.
- \interleave_strings(s1: string, s2: string)\ - Interleave characters from two strings.
- \join_to_string(iterable: string, separator: string =  )\ - Joins an iterable of strings into a single string with a specified separator.
- \kebab_case(text: string)\ - Convert whitespace-separated *text* to kebab-case (lowercase, hyphens).
- \left_bytes(text: string, num_bytes: integer)\ - Return leading characters from *text* measured in UTF-16 LE bytes.
- \left_substring(original_string: string, num_chars: integer)\ - Extrae un número especificado de caracteres del principio de una cadena.
- \longest_common_prefix(s1: string, s2: string)\ - Return the longest common prefix of two strings.
- \longest_common_suffix(s1: string, s2: string)\ - Return the longest common suffix of two strings.
- \longest_word(text: string)\ - Returns the longest word in a text string.
- \metaphone(text: string)\ - Computes the Metaphone phonetic code for a word.
- \mid_bytes(text: string, start: integer, length: integer = None)\ - Return a substring of *text* measured in UTF-16 LE bytes.
- \move_word(input_string: string, from_index: integer, to_index: integer)\ - Moves a word from one position to another within a string.
- \normalize_unicode(text: string, form: string = NFC)\ - Normalize a Unicode string to a canonical form.
- \nysiis(text: string)\ - Compute the NYSIIS (New York State Identification and Intelligence System) phonetic code.
- \pad_center(text: string, width: integer, fillchar: string =  )\ - Centre *text* within *width* characters, padding both sides.
- \pad_end(text: string, width: integer, fillchar: string =  )\ - Pad *text* on the right to reach *width* characters.
- \pad_start(text: string, width: integer, fillchar: string =  )\ - Pad *text* on the left to reach *width* characters.
- \position_in_string(main_string: string, substring: string, start_position: integer = 1)\ - Devuelve una lista de todas las posiciones (basadas en 1) donde se encuentra una subcadena
- \position_in_string_reverse(main_string: string, substring: string, start_position: integer = -1)\ - Devuelve una lista de todas las posiciones (basadas en 1) donde se encuentra una subcadena
- \random_string(length: integer = 10, secure: boolean = True)\ - Generates a random string of a specified length.
- \regex_replace(text: string, pattern: string, replacement: string, case_insensitive: boolean = False)\ - Replaces all occurrences matching a regular expression pattern with a replacement string.
- \remove_blank_lines(text: string)\ - Removes all blank (empty or whitespace-only) lines from a text string.
- \remove_prefix(text: string, prefix: string)\ - Remove *prefix* from the start of *text* if present.
- \remove_suffix(text: string, suffix: string)\ - Remove *suffix* from the end of *text* if present.
- \repeat_each_char(text: string, n: integer)\ - Repeat each character in *text* n times.
- \repeat_string(text: string, times: integer)\ - Repeats a string a given number of times.
- \replace_by_position(original_string: string, start_position: integer, num_chars: integer, new_text: string)\ - Replaces a segment of a string defined by position and length with new text.
- \replace_first_occurrence(text: string, old_substring: string, new_substring: string)\ - Replaces only the first occurrence of a specified substring within a string.
- \replace_last_occurrence(text: string, old_substring: string, new_substring: string)\ - Replaces only the last occurrence of a specified substring within a string.
- \replace_string(original_string: string, old_substring: string, new_substring: string)\ - Reemplaza todas las ocurrencias de una subcadena en una cadena con otra subcadena.
- \reverse_string(input_string: string)\ - Invierte una cadena de texto.
- \reverse_words(text: string)\ - Reverses the order of words in a string.
- \right_bytes(text: string, num_bytes: integer)\ - Return trailing characters from *text* measured in UTF-16 LE bytes.
- \right_substring(original_string: string, num_chars: integer)\ - Extrae un número especificado de caracteres del final de una cadena.
- \rot13(text: string)\ - Apply ROT13 rotation to *text*.
- \rotate_words(text: string, num_rotations: integer, rotate_direction: string)\ - Cyclically rotates words within a text string.
- \run_length_decode(encoded: string)\ - Decode a run-length encoded string.
- \run_length_encode(text: string)\ - Run-length encode a string.
- \search_text(find_text: string, within_text: string, start_num: integer = 1)\ - Case-insensitive text search returning 1-based position.
- \shortest_word(text: string)\ - Returns the shortest word in a text string.
- \slugify(text: string, separator: string = -)\ - Converts text to a URL-friendly slug.
- \snake_to_camel(text: string)\ - Convert snake_case to camelCase.
- \sort_lines(text: string, reverse: boolean = False)\ - Sorts lines in a text string alphabetically.
- \soundex(text: string)\ - Computes the Soundex phonetic code for a word.
- \split_all(text_to_tokenize: string, delimiter_pattern: string = None, return_joined: boolean = False)\ - Splits the input string into a list of words (tokens) based on a
- \split_between_delimiters(text: string, delimiter_key: string)\ - Splits the input string into a list of substrings, using the content within
- \split_by_substrings(p_iparse: string, p_separators: array)\ - Splits a string by a list of keyword separators, keeping the separator with its content.
- \squeeze(text: string, char: string)\ - Reduce consecutive runs of *char* to a single occurrence.
- \string_merge(string1: string, string2: string, base: string = None)\ - Merges two strings based on a common base string (3-way merge).
- \string_xor(a: string, b: string)\ - XOR two equal-length strings character by character.
- \strip_html_tags(text: string)\ - Removes all HTML/XML tags from a string.
- \substitute(text: string, old_text: string, new_text: string, instance_num: integer = 0)\ - Substitutes new text for old text in a string.
- \substring(original_string: string, start_position: integer, length: integer)\ - Extrae una subcadena de una cadena a partir de una posición inicial y con una longitud específica.
- \substring_after_last_digit(input_string: string)\ - Extracts the substring after the last occurrence of a digit in a given string.
- \substring_before_first_digit(input_string: string)\ - Extracts the substring before the first occurrence of a digit in a given string.
- \substring_between_delimiters(text: string, delimiter_key: string)\ - Extracts all substrings found between a specified pair of delimiters.
- \substring_between_pattern(input_string: string, pattern: string)\ - Extracts the substring located between the first two occurrences of a specified pattern
- \substring_from_delimiters(input_string: string, opening_delimiter: string = (, closing_delimiter: string = ))\ - Extracts the substring enclosed within the first pair of specified delimiters in a given string.
- \substring_on_left(text: string, pattern: string)\ - Extracts the substring that appears before the first occurrence of a specified pattern.
- \substring_on_right(text: string, pattern: string)\ - Extracts the substring that appears after the first occurrence of a specified pattern.
- \surround(text: string, wrapper: string)\ - Wraps a text with a given string on both sides.
- \swap_case(text: string)\ - Swap the case of every character in *text*.
- \text_after(text: string, delimiter: string, instance_num: integer = 1)\ - Returns text that occurs after a given delimiter.
- \text_before(text: string, delimiter: string, instance_num: integer = 1)\ - Returns text that occurs before a given delimiter.
- \text_split(text: string, col_delimiter: string = None, row_delimiter: string = None)\ - Splits text into rows and/or columns using delimiters.
- \title_case(text: string)\ - Convert *text* to title case, capitalising the first letter of each word.
- \truncate_string(input_string: string, target_length: integer = 255, align: string = right)\ - Truncates a string to a specified target length from the left or right.
- \truncate_with_ellipsis(text: string, max_len: integer, ellipsis: string = ...)\ - Truncate *text* to *max_len* characters, appending *ellipsis* if cut.
- \word_at(text: string, index: integer)\ - Return the word at the given 1-indexed position.
- \wrap_text(text: string, width: integer = 80)\ - Wraps text to a specified line width at word boundaries.
- \zigzag_case(text: string)\ - Alternate character case: even indices lower, odd indices upper.

## fxString.string_regex

- \regex_count(text: string, pattern: string, case_sensitive: boolean = True)\ - Counts the number of non-overlapping matches of a regex pattern.
- \regex_extract(text: string, pattern: string, group: integer = 0, case_sensitive: boolean = True)\ - Extracts the first match of a regex pattern from a string.
- \regex_extract_all(text: string, pattern: string, case_sensitive: boolean = True)\ - Extracts all matches of a regex pattern from a string.
- \regex_match(text: string, pattern: string, case_sensitive: boolean = True)\ - Checks whether a string matches a regular expression pattern.
- \regex_split(text: string, pattern: string, max_split: integer = 0, case_sensitive: boolean = True)\ - Splits a string by a regex pattern.

## fxString.string_similarity

- \are_words_equivalent(word1: string, word2: string, levenshtein_threshold: number = 0.85, jaro_winkler_threshold: number = 0.9, metaphone_required: boolean = True)\ - Propuesta de mejores prácticas para determinar si dos palabras son
- \calculate_similarity(word1: string, word2: string, algorithm: any = levenshtein, kwargs: any)\ - Función wrapper unificada para calcular similitud entre dos palabras/frases
- \find_closest_match(target: string, candidates: array, algorithm: string = levenshtein)\ - Finds the best matching string from a list of candidates.
- \find_common_words(string_one: string, string_two: string)\ - Identifies and returns the common unique words between two input strings.
- \generate_ngrams(text: string, n: integer = 2)\ - Generates character n-grams from a string.
- \has_same_characters(string_one: string, string_two: string)\ - Checks if two strings contain exactly the same unique characters,
- \has_same_words(string_one: string, string_two: string)\ - Checks if two strings contain the same words, ignoring word order and case.
- \rank_by_similarity(target: string, candidates: array, algorithm: string = levenshtein, top_n: integer = 0)\ - Ranks a list of candidates by similarity to a target string.
- \string_cosine_score(a: string, b: string, n: integer = 2)\ - Calculates cosine similarity between two strings using character n-grams.
- \string_diflib_seqmatch_score(a: string, b: string)\ - Calculates the similarity score using difflib.SequenceMatcher.
- \string_dna_score(a: string, b: string)\ - Calculates the Needleman-Wunsch similarity scores (DNA sequence alignment).
- \string_hamming_score(a: string, b: string)\ - Calculates the Hamming similarity scores between two strings.
- \string_jaccard_score(a: string, b: string)\ - Calculates the Jaccard similarity index based on tokens (words).
- \string_jarowinkler_score(a: string, b: string)\ - Calculates the Jaro-Winkler similarity scores.
- \string_lcs_record(a: string, b: string)\ - Calculates the Longest Common Subsequence (LCS) score and extracts the actual LCS string.
- \string_lcs_score(a: string, b: string)\ - Calculates the Longest Common Subsequence (LCS) similarity score.
- \string_levenshtein_score(a: string, b: string)\ - Calculates the Levenshtein (edit) distance and similarity scores.
- \string_mra_score(a: string, b: string)\ - Calculates the Match Rating Approach (MRA) similarity scores.
- \string_ratcliffobershelp_score(a: string, b: string)\ - Calculates the Ratcliff-Obershelp pattern recognition similarity scores.
- \string_similarity_score(a: string, b: string)\ - Calculates similarity scores using multiple string comparison algorithms.
- \string_sorensendice_score(a: string, b: string)\ - Calculates the Sørensen-Dice similarity coefficient based on tokens (words).

## fxString.string_spanish

- \fix_spanish(input_string: string, additional_allowed_chars: string = )\ - Cleans an input string by replacing specific characters and filtering
- \is_valid_cif(cif_value: string)\ - Validates a Spanish CIF (Código de Identificación Fiscal) format and control character.
- \is_valid_dni(dni_value: string)\ - Validates a Spanish DNI (Documento Nacional de Identidad) format and control letter.
- \is_valid_nie(nie_value: string)\ - Validates a Spanish NIE (Número de Identificación de Extranjero) format and control letter.
- \nif_letter(p_dni: string)\ - Calculates and appends the control letter to a Spanish DNI/NIE numeric part.
- \nif_padding(p_nif: string)\ - Attempts to format an incomplete Spanish identification number by padding with zeros.
- \nif_parse(nif: string)\ - Validates if a Spanish identification number has the correct format.
- \reduce_spanish_letters(input_string: string, strength: integer)\ - Reduces phonetic and orthographic variations in Spanish strings based on a
- \remove_spanish_stop_words(input_string: string)\ - Removes common Spanish articles, conjunctions, and prepositions (stop words)
- \validate_spanish_nif(nif_type: string, nif_value: string)\ - Validates a Spanish NIF (DNI, NIE, or CIF) based on its type and value.

## fxString.string_validations

- \contains_digit(input_string: string)\ - Checks if the given string contains at least one digit.
- \same_letters(string_a: string, string_b: string)\ - Checks if two strings contain the exact same characters, irrespective of their order.

## fxVBA.array_functions

- \Array_(values: array = ...)\ - Description
- \Filter_(source_array: array, match: string, include: boolean = True, compare: integer = 0)\ - Description
- \Join_(source_array: array, delimiter: string =  )\ - Description
- \LBound(array_var: array, dimension: integer = 1)\ - Description
- \Split(expression: string, delimiter: string =  , limit: integer = -1, compare: integer = 0)\ - Description
- \UBound(array_var: array, dimension: integer = 1)\ - Description

## fxVBA.conversion_functions

- \CBool(expression: any)\ - Description
- \CByte(expression: any)\ - Description
- \CCur(expression: any)\ - Description
- \CDate(expression: any)\ - Description
- \CDbl(expression: any)\ - Description
- \CDec(expression: any)\ - Description
- \CInt(expression: any)\ - Description
- \CLng(expression: any)\ - Description
- \CLngLng(expression: any)\ - Description
- \CLngPtr(expression: any)\ - Description
- \CSng(expression: any)\ - Description
- \CStr(expression: any)\ - Description
- \CVErr(error_number: integer)\ - Description
- \CVar(expression: any)\ - Description
- \DateValue(date_str: string)\ - Description
- \TimeValue(time_str: string)\ - Description
- \Val(string: string)\ - Description

## fxVBA.date_functions

- \DateAdd(interval: string, number: number, date_val: any)\ - Description
- \DateDiff(interval: string, date1: any, date2: any, first_day_of_week: integer = 0, first_week_of_year: integer = 0)\ - Description
- \DatePart(interval: string, date_val: any, first_day_of_week: integer = 0, first_week_of_year: integer = 0)\ - Description
- \DateSerial(year: integer, month: integer, day: integer)\ - Description
- \Date_()\ - Description
- \Day(date_val: any)\ - Description
- \Hour(time_val: any)\ - Description
- \Minute(time_val: any)\ - Description
- \Month(date_val: any)\ - Description
- \MonthName(month_num: integer, abbreviate: boolean = False)\ - Description
- \Now()\ - Description
- \Second(time_val: any)\ - Description
- \TimeSerial(hour: integer, minute: integer, second: integer)\ - Description
- \Time_()\ - Description
- \Timer()\ - Description
- \WeekDay(date_val: any, first_day_of_week: integer = 1)\ - Description
- \WeekDayName(weekday_num: integer, abbreviate: boolean = False, first_day_of_week: integer = 1)\ - Description
- \Year(date_val: any)\ - Description

## fxVBA.financial_functions

- \DDB(cost: number, salvage: number, life: number, period: number, factor: number = 2.0)\ - Description
- \FV(rate: number, nper: number, pmt: number, pv: number = 0, type_: integer = 0)\ - Description
- \IPmt(rate: number, per: number, nper: number, pv: number, fv: number = 0, type_: integer = 0)\ - Description
- \IRR(values: array, guess: number = 0.1)\ - Description
- \MIRR(values: array, finance_rate: number, reinvest_rate: number)\ - Description
- \NPV(rate: number, values: array)\ - Description
- \NPer(rate: number, pmt: number, pv: number, fv: number = 0.0, type_: integer = 0)\ - Description
- \PPmt(rate: number, per: number, nper: number, pv: number, fv: number = 0, type_: integer = 0)\ - Description
- \PV(rate: number, nper: number, pmt_: number, fv: number = 0, type_: integer = 0)\ - Description
- \Pmt(rate: number, nper: number, pv: number, fv: number = 0, type_: integer = 0)\ - Description
- \Rate(nper: number, pmt_: number, pv: number, fv: number = 0, type_: integer = 0, guess: number = 0.1)\ - Description
- \SLN(cost: number, salvage: number, life: number)\ - Description
- \SYD(cost: number, salvage: number, life: number, period: number)\ - Description

## fxVBA.format_functions

- \FormatCurrency(expression: number, num_decimals: integer = 2, include_leading_digit: integer = -2, use_parens_for_negative: integer = -2, group_digits: integer = -2)\ - Description
- \FormatDateTime(expression: any, named_format: integer = 0)\ - Description
- \FormatNumber(expression: number, num_decimals: integer = 2, include_leading_digit: integer = -2, use_parens_for_negative: integer = -2, group_digits: integer = -2)\ - Description
- \FormatPercent(expression: number, num_decimals: integer = 2, include_leading_digit: integer = -2, use_parens_for_negative: integer = -2, group_digits: integer = -2)\ - Description
- \Format_(expression: any, format_str: string = , first_day_of_week: integer = 0, first_week_of_year: integer = 0)\ - Description

## fxVBA.logic_functions

- \Choose(index_num: integer, values: array = ...)\ - Description
- \IIf(expression: boolean, true_part: any, false_part: any)\ - Description
- \IsArray(variable: any)\ - Description
- \IsDate(expression: any)\ - Description
- \IsEmpty(variable: any)\ - Description
- \IsError(number: any)\ - Description
- \IsMissing(argument: any)\ - Description
- \IsNull(expression: any)\ - Description
- \IsNumeric(expression: any)\ - Description
- \IsObject(variable: any)\ - Description
- \Switch(args: array = ...)\ - Description

## fxVBA.math_functions

- \Abs_(number: number)\ - Description
- \Atn(number: number)\ - Description
- \Cos(number: number)\ - Description
- \Exp(number: number)\ - Description
- \Fix(number: number)\ - Description
- \Int_(number: number)\ - Description
- \Log(number: number)\ - Description
- \Rnd(number: integer = None)\ - Description
- \Round_(number: number, num_digits_after_decimal: integer = 0)\ - Description
- \Sgn(number: number)\ - Description
- \Sin(number: number)\ - Description
- \Sqr(number: number)\ - Description
- \Tan(number: number)\ - Description

## fxVBA.misc_functions

- \AccessError(error_number: integer)\ - Description
- \CurrentUser()\ - Description
- \Eval_(expression: string)\ - Description
- \Hex_(number: integer)\ - Description
- \Oct_(number: integer)\ - Description
- \QBColor(color: integer)\ - Description
- \RGB(red: integer, green: integer, blue: integer)\ - Description
- \SysCmd(action: integer, argument2: integer = None, argument3: string = None)\ - Description

## fxVBA.string_functions

- \Asc(string: string)\ - Description
- \AscB(string: string)\ - Description
- \ChrW(char_code: integer)\ - Description
- \Chr_(char_code: integer)\ - Description
- \InStr(start: integer = 1, string1: string = , string2: string = , compare: integer = 0)\ - Description
- \InStrRev(string1: string, string2: string, start: integer = -1, compare: integer = 0)\ - Description
- \LCase(string: string)\ - Description
- \LTrim(string: string)\ - Description
- \Left(string: string, length: integer)\ - Description
- \LeftB(string: string, length: integer)\ - Description
- \LenB(expression: string)\ - Description
- \Len_(expression: string)\ - Description
- \Mid(string: string, start: integer, length: integer = None)\ - Description
- \MidB(string: string, start: integer, length: integer = None)\ - Description
- \RTrim(string: string)\ - Description
- \Replace(expression: string, find: string, replace_with: string, start: integer = 1, count: integer = -1, compare: integer = 0)\ - Description
- \Right(string: string, length: integer)\ - Description
- \RightB(string: string, length: integer)\ - Description
- \Space(number: integer)\ - Description
- \StrComp(string1: string, string2: string, compare: integer = 0)\ - Description
- \StrConv(string: string, conversion: integer, lcid: integer = None)\ - Description
- \StrReverse(expression: string)\ - Description
- \Str_(number: number)\ - Description
- \String(number: integer, character: string)\ - Description
- \Trim(string: string)\ - Description
- \UCase(string: string)\ - Description

## fxVBA.system_functions

- \BuildCriteria(field: string, field_type: integer, expression: string)\ - Description
- \Command()\ - Description
- \CurDir(drive: string = None)\ - Description
- \Dir_(pathname: string = , attributes: integer = 0)\ - Description
- \Environ(expression: string)\ - Description
- \Error(err_number: integer)\ - Description
- \FileDateTime(pathname: string)\ - Description
- \FileLen(pathname: string)\ - Description
- \GetAttr_(pathname: string)\ - Description
- \Nz(value: any, value_null: any = )\ - Description
- \Partition(number: number, start: number, stop: number, interval: number)\ - Description
- \TypeName(value: any)\ - Description
- \VarType(value: any)\ - Description
