complex op• 資料種類:roots × roots → cimage
• 呼叫: import fullseye as fs; fs.ledger.cplx_rational_field(zeros, poles, gain=1.0, centre=0j, half_width=2.0, shape=(256, 256))(要直接呼叫實作,import mathops; mathops.cplx_rational_field(zeros, poles, gain=1.0, centre=0j, half_width=2.0, shape=(256, 256));從台帳取用則 opsmath.get("cplx_rational_field"))
對有理函數 `R(z) = gain * prod(z - zeros) / prod(z - poles)` 取樣。
> 以下的詳細說明為原文 —— 摘要與標題已翻譯。
Returns the complex image `R(z)` over a rectangular window of the plane —
the *area* counterpart of this family's contour operators. The product is
formed in log space (`exp(sum(log(z - zk)) - sum(log(z - pm)))`) so a
degree-30 numerator does not overflow before the denominator can divide it
back down; the branch cuts of the individual logarithms cancel in the
exponential, so the result is the ordinary principal value of the quotient,
not a branch of it.
★Why this earns its place: the field is not decoration. The argument
principle says that the winding number of `R` along a closed contour
equals *(zeros inside) - (poles inside)*, counted with multiplicity — so
this family's own `cplx_argument_principle / cplx_winding_number`
are an independent oracle for every field this operator produces, and
:func:cplx_domain_colour makes that integer visible as the number of
times the hue cycles.
Parameters
----------
zeros, poles : complex array-like
Roots of the numerator and denominator, repeated for multiplicity.
Either may be empty (a polynomial, or `1/q). poly_roots` produces
exactly this `roots` vocabulary.
gain : complex
Leading coefficient.
centre, half_width, shape :
The window, as in :func:cplx_plane_grid.
Raises `ValueError`: a pole (or zero) lands *exactly* on a sample, so
the value there is not a number — nudge `centre` by half a pixel or take
an odd `shape` (the message says which pole and where); non-finite input;
the window is degenerate; the result overflows to infinity anyway (the gain
and the window disagree by more than float64 can hold).
HALCON: no operator (HALCON has no complex-plane family).
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。
• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py
cimage 作為輸入)cplx_cr_residual · cplx_domain_colour · mandelbrot_interior
complex)cplx_contour_circle · cplx_poly_eval · cplx_contour_integral · cplx_winding_number · cplx_cauchy_value · cplx_argument_principle · cplx_laurent_coeffs · cplx_joukowski
*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.