complex oproots × roots → cimageimport fullseye as fs; fs.ledger.cplx_rational_field(zeros, poles, gain=1.0, centre=0j, half_width=2.0, shape=(256, 256)) (実装を直接呼ぶなら import mathops; mathops.cplx_rational_field(zeros, poles, gain=1.0, centre=0j, half_width=2.0, shape=(256, 256))、台帳から引くなら opsmath.get("cplx_rational_field"))Sample a rational function R(z) = gain * prod(z - zeros) / prod(z - poles).
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.
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 lands exactly on a sample, so the value
there is not finite — nudge centre by half a pixel or change shape
(the message says which pole and where). A sampled zero is returned as zero.
Also raises for 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).
Empty input (zeros with 0 elements) returns the poles-only field with
the same shape as z (reason: an empty product is the multiplicative identity).
mathops の全 op は入力を検証してから計算する(黙って通さない):
ValueError — float64 への強制変換は虚部を黙って捨てる(numpy は ComplexWarning だけ出して「もっともらしく間違った」実数を返す)。.real/.imag/abs() を明示するか、複素対応の complexops を使う。ValueError — マスクを剥がして下の生値を使う暗黙変換を拒否。埋める/落とすを明示する。ValueError(件数を明示して拒否 — 結果全体に伝播するため)。ValueError。reshape を明示する)。stat_histogram の bins は mathops.MAX_ELEMENTS(2^26 ≈ 6700 万要素)超で ValueError。py -3.11 examples/poc_complex_plane_fields.pycimage を入力に取れる)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 operator registry. この per-op ノートは tools/opdocs.py md が自動生成(手編集しない)。
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.