complex op• Data kinds: cpoints → cpoints
• Call: import mathops; mathops.cplx_joukowski(z, c=1.0) (or opsmath.get("cplx_joukowski"))
Joukowski (Zhukovsky) conformal map `w = z + c^2 / z`.
The classical aerofoil map (Zhukovsky 1910): the circle `|z| = c` folds
onto the flat plate `[-2c, 2c] of the real axis (z = c e^(i t)` gives
`w = 2c cos t` — exact, and what the tests pin); a circle of radius
`R > c` centred at the origin maps to the ellipse with semi-axes
`R + c^2/R and R - c^2/R; and a circle through z = c` whose
centre is offset into the second quadrant maps to a cambered aerofoil with
a cusped trailing edge — the reason the map exists.
Conformal (angle-preserving) everywhere except at `z = ±c`, where the
derivative `1 - c^2/z^2` vanishes and angles are doubled — that is
what creates the cusp, and it is a property of the map, not a defect.
Raises `ValueError: a sample at z = 0` (the map's pole), a result
that overflowed (a sample so close to 0 that `c^2/z` leaves float64
range), a non-finite or non-positive-real *c*, plus the usual shape and
finiteness contracts.
HALCON: no operator (conformal maps are not part of its transform set).
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
cpoints as input)cplx_poly_eval · cplx_contour_integral · cplx_winding_number · cplx_cauchy_value · cplx_argument_principle · cplx_laurent_coeffs · cplx_mobius
complex)cplx_contour_circle · cplx_poly_eval · cplx_contour_integral · cplx_winding_number · cplx_cauchy_value · cplx_argument_principle · cplx_laurent_coeffs · 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.