complex opcpoints × cpoints → cscalarimport fullseye as fs; fs.ledger.cplx_cauchy_value(z, fz, w) (実装を直接呼ぶなら import mathops; mathops.cplx_cauchy_value(z, fz, w)、台帳から引くなら opsmath.get("cplx_cauchy_value"))Cauchy’s integral formula: recover f(w) inside a contour from its
values on the contour.
f(w) = 1/(2*pi*i*n) ∮ f(zeta)/(zeta - w) dzeta where n is the
winding number of the contour around w (Cauchy 1831; the division by
n is what makes a doubly-wound contour give the same answer). Valid
only if f is holomorphic on and inside the contour — nothing here can
check that, and this is the honest limit of the op: fed values of a
non-holomorphic f (or of one with a pole inside) it returns the
integral, which is then simply not f(w).
Accuracy inherits the O(n^-2) chordal quadrature of
:func:cplx_contour_integral and degrades as w approaches the path
(the integrand’s peak sharpens): measured for f(z) = z**2 on a
256-point unit circle, the absolute error is 9.0e-6 at w = 0.3 and
8.1e-5 at w = 0.9 — 9x worse for a point 7x closer to the path
(0.7 -> 0.1 of clearance). The blow-up is real but gradual; what it does
not survive is clearance below one sampling step, which is refused.
Raises ValueError: w outside the contour (winding 0 — the
integral is then 0 and returning it as “f(w)” would be a lie), w closer
to the contour than one sampling step (the quadrature is meaningless
there — refine the contour), plus everything
:func:cplx_winding_number and :func:cplx_contour_integral refuse.
HALCON: no operator.
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/math_complex.pycscalar を入力に取れる)—
complex)cplx_contour_circle · cplx_poly_eval · cplx_contour_integral · cplx_winding_number · cplx_argument_principle · cplx_laurent_coeffs · cplx_joukowski · cplx_mobius
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.