complex op• 데이터 종류: cpoints × cpoints → table
• 호출: import mathops; mathops.cplx_laurent_coeffs(z, fz, kmin=-1, kmax=4)(또는 opsmath.get("cplx_laurent_coeffs"))
균일하게 표본화한 원 위의 로랑(및 테일러) 계수 —— 유수 포함.
> 아래 상세 설명은 원문입니다 —— 요약과 제목은 번역되어 있습니다.
For `f holomorphic on an annulus around c`,
`f(z) = sum_k c_k (z - c)^k` with
`c_k = 1/(2*pi*i) ∮ f(zeta)/(zeta - c)^(k+1) dzeta`. On a circle of
radius `r sampled at n` equally spaced angles this becomes a discrete
Fourier sum, `c_k = (1/(n r^k)) sum_j f_j exp(-i k theta_j)` — the
trapezoidal rule in the angle, where it converges geometrically rather
than as `O(n^-2)` (Trefethen & Weideman 2014, "The exponentially
convergent trapezoidal rule").
`c_-1 **is the residue** at c (when c` is the only singularity
inside), `c_k for k >= 0` are the Taylor coefficients
`f^(k)(c)/k!, and a non-zero c_-m for m > 1` reveals a pole of
order `m. Measured on the unit circle with f = 1/(z - 0.5)`,
`n = 64: c_-1 = 1 and c_-2 = 0.5` to 1e-16 (machine precision).
Returns a dict: `k (int64 orders, kmin..kmax) · c` (complex128
coefficients) · `center · radius`. The centre is the sample mean,
which is exact for a uniformly sampled circle.
Orientation, and how it differs from the rest of the family: the sum
runs over the sample *set*, not the sample *order*, so this op always
returns the coefficients of the positively oriented circle — the standard
definition — whatever order the points arrive in. Feed a clockwise circle
and `c_-1 still comes back +` the residue, while
`cplx_contour_integral / (2*pi*i) on the same points returns -` it
(verified). Both are right; they answer different questions (the intrinsic
coefficient vs the integral along *this* traversal). Do not cross-check one
against the other without fixing the orientation first.
Honest limitation — aliasing: the discrete sum cannot distinguish
`c_k from c_{k+n}`, so a coefficient carries the alias sum
`sum_m c_{k+m n} r^{m n}`. That is negligible for a rapidly converging
series (the `0.5^64` term above) and ruinous near the annulus boundary.
Requesting more than `n` coefficients is refused for the same reason.
Raises `ValueError`: the samples are not a uniformly spaced circle
(unequal radii or unequal angular gaps beyond `1e-8` relative — this op
is *not* valid on an arbitrary contour, and silently pretending otherwise
would return numbers that mean nothing), `kmin > kmax, more than n`
coefficients requested, non-integer orders, and a coefficient that
overflowed (`r^-k` for a small radius and a large negative order).
HALCON: no operator.
mathops 의 모든 연산자는 입력을 검증한 뒤에 계산합니다(조용히 통과시키지 않습니다):
• **complex 입력은 ValueError** —— float64 로의 강제 변환은 허수부를 조용히 버립니다(numpy 는 ComplexWarning 만 내고 「그럴듯하게 틀린」 실수를 돌려줍니다). .real/.imag/abs() 를 명시하거나 복소수를 다루는 complexops 를 쓰세요.
• **masked 요소가 있는 masked array 는 ValueError** —— 마스크를 벗겨 아래 원값을 쓰는 암묵 변환을 거부합니다. 채울지 버릴지를 명시하세요.
• **NaN/Inf 는 모든 입력에서 ValueError**(개수를 명시하고 거부 —— 결과 전체로 전파되므로).
• 형상은 엄격: 1-D 와 2-D 를 암묵적으로 승격·브로드캐스트하지 않습니다(vector 슬롯에 matrix, matrix 슬롯에 vector 는 ValueError. reshape 를 명시하세요).
• 크기 상한: 행렬을 받는 연산자와 stat_histogram 의 bins 는 mathops.MAX_ELEMENTS(2^26 ≈ 6700 만 요소)를 넘으면 ValueError.
• 샘플 데이터 카탈로그(DL URL / 라이선스) —— 2-D 는 skimage.data(BSD/public)+ 합성, 3-D 는 실데이터 소스(Stanford/PDS 등)의 DL URL.
• 연산자의 내력·참고문헌 —— 이 연산자 족의 바탕이 된 연구/기법의 출처.
• 알고리즘의 정전(저자·연도)과 용도는 위의 패밀리 사용 가이드에 적혀 있습니다.
• math_complex — py -3.11 examples/math_complex.py
table 를 입력으로 받는 것)—
complex)cplx_contour_circle · cplx_poly_eval · cplx_contour_integral · cplx_winding_number · cplx_cauchy_value · cplx_argument_principle · cplx_joukowski · cplx_mobius
*Provenance: mathops.py — MATH 연산자 레지스트리. 이 op 노트는 tools/opdocs.py md 가 자동 생성합니다(직접 편집하지 마세요).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.