complex op• 資料種類:cpoints × cpoints → measurement
• 呼叫:import mathops; mathops.cplx_argument_principle(z, fz)(或 opsmath.get("cplx_argument_principle"))
輻角原理:僅憑 `f` 的取樣值,數出圍道內零點數減去極點數。
> 以下的詳細說明為原文 —— 摘要與標題已翻譯。
`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 的每個運算子都先檢驗輸入再計算(不讓任何東西無聲通過):
• **complex 輸入一律 ValueError** —— 強制轉成 float64 會無聲丟掉虛部(numpy 只發一個 ComplexWarning,然後回傳一個「看似合理卻是錯的」實數)。請明確寫出 .real/.imag/abs(),或改用支援複數的 complexops。
• **含被遮罩元素的 masked array 一律 ValueError** —— 拒絕「剝掉遮罩直接使用下面原值」的隱式轉換。請明確選擇填補還是丟棄。
• **所有輸入中的 NaN/Inf 一律 ValueError**(明確給出個數後拒絕 —— 它會汙染整個結果)。
• 形狀嚴格:不對 1-D 與 2-D 做隱式提升或廣播(向量槽位收到矩陣、矩陣槽位收到向量都是 ValueError;請明確 reshape)。
• 尺寸上限:接受矩陣的運算子與 stat_histogram 的 bins,超過 mathops.MAX_ELEMENTS(2^26 ≈ 6700 萬個元素)即 ValueError。
• 範例資料目錄(下載 URL / 授權) —— 2-D 用 skimage.data(BSD/公有領域)加合成圖,3-D 給出真實資料源(Stanford/PDS 等)的下載 URL。
• 運算子來歷與參考文獻 —— 該運算子族所依據的研究/方法出處。
• 演算法的正典(作者・年份)與用途見上面的族使用指南。
• math_complex — py -3.11 examples/math_complex.py
measurement 作為輸入)—
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 運算子登記表。本條目由 tools/opdocs.py md 自動產生(請勿手動編輯)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.