complex op• Data kinds: cpoints × cpoints → cscalar
• Call: import fullseye as fs; fs.ledger.cplx_cauchy_value(z, fz, w) (to call the implementation directly, import mathops; mathops.cplx_cauchy_value(z, fz, w); from the registry, 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 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 cplx_winding_number and cplx_contour_integral refuse.
HALCON: no operator.
Every mathops op validates its input before computing (nothing slips through silently):
• complex input raises ValueError — coercing to float64 silently discards the imaginary part (numpy only emits a ComplexWarning and returns a plausible-looking wrong real number). State .real/.imag/abs() explicitly, or use complexops, which handles complex data.
• masked arrays with masked elements raise ValueError — the implicit conversion that peels off the mask and uses the raw values underneath is refused. Say explicitly whether to fill or to drop.
• NaN/Inf raises ValueError on every input (refused with the count stated — it propagates through the whole result).
• Shapes are strict: 1-D and 2-D are never implicitly promoted or broadcast (a matrix in a vector slot, or a vector in a matrix slot, raises ValueError; reshape explicitly).
• Size cap: ops that take a matrix, and the stat_histogram bins, raise ValueError beyond mathops.MAX_ELEMENTS (2^26 ≈ 67 million elements).
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• math_complex — py -3.11 examples/math_complex.py
cscalar as input)—
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. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.