complex op• Data kinds: cpoints × cpoints → measurement
• Call: import fullseye as fs; fs.ledger.cplx_argument_principle(z, fz) (to call the implementation directly, import mathops; mathops.cplx_argument_principle(z, fz); from the registry, opsmath.get("cplx_argument_principle"))
Argument principle: count zeros minus poles enclosed by a contour, from sampled values of f alone.
Z - P = 1/(2*pi*i) ∮ f'/f dz equals the winding number of the image curve f(z) around the origin (Cauchy 1831 / Riemann): as the contour is traversed once counter-clockwise, the argument of f increases by 2*pi (Z - P), counting multiplicities. Computing it as a winding number of the image needs no derivative and no root finding — only f sampled on the path — and returns an exact integer.
Honest limitations, all of them real:
• It returns the difference Z - P, never the two separately. A simple zero and a simple pole inside cancel to 0.
• The result is multiplied by the winding number of the contour itself, so it equals Z - P only for a simple, positively-oriented contour (a clockwise one returns -(Z - P)).
• It is the winding of the *sampled* image polygon, so it aliases low on a coarse contour. A half-turn jump between samples is detected and raised; anything below that is indistinguishable from a genuine slower turn — f = z**5 on a 4-point circle returns 1, not 5 (measured). A RuntimeWarning fires from pi/2 per step onward (WIND_ALIAS_WARN); the verification that actually works is to double n until the count repeats.
Raises ValueError: f vanishes at a sample point (a zero *on* the path — the count is undefined there), the image curve is undersampled (a half-turn between consecutive samples: refine the contour), plus the usual shape/finiteness contracts.
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
measurement as input)—
complex)cplx_contour_circle · cplx_poly_eval · cplx_contour_integral · cplx_winding_number · cplx_cauchy_value · 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.