complex op• 数据种类:cpoints × cpoints → table
• 调用: import fullseye as fs; fs.ledger.cplx_laurent_coeffs(z, fz, kmin=-1, kmax=4)(要直接调用实现,import mathops; mathops.cplx_laurent_coeffs(z, fz, kmin=-1, kmax=4);从台账取用则 opsmath.get("cplx_laurent_coeffs"))
均匀采样圆周上的 Laurent(以及 Taylor)系数 —— 含留数。
> 以下的详细说明为原文 —— 摘要与标题已翻译。
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 的每个算子都先校验输入再计算(不让任何东西无声通过):
• complex 输入一律 ValueError —— 强制转成 float64 会无声丢掉虚部(numpy 只发一个 ComplexWarning,然后返回一个「看着合理却是错的」实数)。请显式写出 .real/.imag/abs(),或改用支持复数的 complexops。
• 含被掩元素的 masked array 一律 ValueError —— 拒绝「剥掉掩码直接使用下面原值」的隐式转换。请显式选择填充还是丢弃。
• 所有输入中的 NaN/Inf 一律 ValueError(明确给出个数后拒绝 —— 它会污染整个结果)。
• 形状严格:不对 1-D 与 2-D 做隐式提升或广播(向量槽位收到矩阵、矩阵槽位收到向量都是 ValueError;请显式 reshape)。
• 尺寸上限:接受矩阵的算子与 stat_histogram 的 bins,超过 mathops.MAX_ELEMENTS(2^26 ≈ 6700 万个元素)即 ValueError。
• 示例数据目录(下载 URL / 许可证) —— 2-D 用 skimage.data(BSD/公有领域)加合成图,3-D 给出真实数据源(Stanford/PDS 等)的下载 URL。
• 算子来历与参考文献 —— 该算子族所依据的研究/方法出处。
• 算法的正典(作者、年份)与用途见上面的族使用指南。
• math_complex — py -3.11 examples/math_complex.py
table 作为输入)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 算子登记表。本条目由 tools/opdocs.py md 自动生成(请勿手工编辑)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.