complex op• 데이터 종류: cpoints × cpoints → table
• 호출: import fullseye as fs; fs.ledger.cplx_laurent_coeffs(z, fz, kmin=-1, kmax=4, method='exact')(구현을 직접 호출하려면 import mathops; mathops.cplx_laurent_coeffs(z, fz, kmin=-1, kmax=4, method='exact'), 원장에서 가져오려면 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).
method="exact" (default) preserves the direct Fourier sum. method="fast" uses an FFT only when angular gaps agree within 64 * float64 epsilon; otherwise it routes to the exact sum. On 4,096 samples and 201 orders the fast path measured over 15 times faster, while differing from the direct sum by less than 1e-12.
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)dynsys_poincare_section · dwt_inverse · chebyshev_eval_nd
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.