complex op• Data kinds: cimage → rgbimage
• Call: import fullseye as fs; fs.ledger.cplx_domain_colour(field, gamma=1.0, bands=0.0, saturation=1.0) (to call the implementation directly, import mathops; mathops.cplx_domain_colour(field, gamma=1.0, bands=0.0, saturation=1.0); from the registry, opsmath.get("cplx_domain_colour"))
Domain colouring: turn a complex field into an RGB image you can read.
Hue carries arg(z) (one full turn of the colour wheel per turn of the argument, red at arg = 0); value carries |z| through the strictly increasing map t/(1+t) with t = |z|**gamma, so a zero is exactly black, |z| = 1 is half brightness and a large modulus saturates at full brightness. The saturation is not dropped far from the origin, so the hue — and with it the argument — stays readable everywhere; pass saturation = 0 for a plain grey ramp of the modulus alone. With bands = 0 (the default) the brightness is monotone in |z|, which means the picture is *invertible*:
the argument comes back out of the hue and the modulus out of the value.
★The picture proves a theorem. Walk a small circle around a zero of order *m* and the hue cycles through the colour wheel exactly *m* times; around a pole of order *m*, *m* times the other way. That is the argument principle, read off the image with no numbers — and this family's cplx_winding_number counts the same integer from the field itself, so the drawing and the arithmetic check each other.
field : complex 2-D array (cimage)
Typically from cplx_rational_field.
gamma : float > 0
Compresses (<1) or stretches (>1) the modulus ramp.
bands : float >= 0
Classic modulus contours: bands shading cycles per decade of
|z|. Non-zero breaks monotonicity (that is the point — it draws
level lines), so the inverse-mapping guarantee above holds only at 0.
saturation : float in [0, 1]
Raises ValueError: not a 2-D complex array; non-finite samples (a pole sampled exactly — see cplx_rational_field); gamma <= 0; bands < 0; saturation outside [0, 1].
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.
• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py
rgbimage as input)—
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. 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.