complex op• 数据种类:roots × roots → cimage
• 调用: import fullseye as fs; fs.ledger.cplx_rational_field(zeros, poles, gain=1.0, centre=0j, half_width=2.0, shape=(256, 256))(要直接调用实现,import mathops; mathops.cplx_rational_field(zeros, poles, gain=1.0, centre=0j, half_width=2.0, shape=(256, 256));从台账取用则 opsmath.get("cplx_rational_field"))
对有理函数 `R(z) = gain * prod(z - zeros) / prod(z - poles)` 采样。
> 以下的详细说明为原文 —— 摘要与标题已翻译。
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。
• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py
cimage 作为输入)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 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.