complex op• 数据种类:cpoints × cpoints → measurement
• 调用:import mathops; mathops.cplx_argument_principle(z, fz)(或 opsmath.get("cplx_argument_principle"))
辐角原理:仅凭 `f` 的采样值,数出围道内零点数减去极点数。
> 以下的详细说明为原文 —— 摘要与标题已翻译。
`Z - P = 1/(2*pi*i) ∮ f'/f dz` equals the winding number of the **image
curve** `f(z)` around the origin (Cauchy 1831 / Riemann): as the contour
is traversed once counter-clockwise, the argument of `f` increases by
`2*pi (Z - P)`, counting multiplicities. Computing it as a winding number
of the image needs no derivative and no root finding — only `f` sampled
on the path — and returns an exact integer.
Honest limitations, all of them real:
• It returns the difference `Z - P`, never the two separately. A
simple zero and a simple pole inside cancel to 0.
• The result is multiplied by the winding number of the contour itself,
so it equals `Z - P` only for a simple, positively-oriented
contour (a clockwise one returns `-(Z - P)`).
• It is the winding of the *sampled* image polygon, so it aliases low
on a coarse contour. A half-turn jump between samples is detected and
raised; anything below that is indistinguishable from a genuine slower
turn — `f = z**5` on a 4-point circle returns 1, not 5 (measured).
A `RuntimeWarning fires from pi/2` per step onward
(:data:WIND_ALIAS_WARN); the verification that actually works is to
double `n` until the count repeats.
Raises `ValueError: f` vanishes at a sample point (a zero *on*
the path — the count is undefined there), the image curve is undersampled
(a half-turn between consecutive samples: refine the contour), plus the
usual shape/finiteness contracts.
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
measurement 作为输入)—
complex)cplx_contour_circle · cplx_poly_eval · cplx_contour_integral · cplx_winding_number · cplx_cauchy_value · cplx_laurent_coeffs · 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.