complex op• Data kinds: cpoints × cpoints → table
• Call: import mathops; mathops.cplx_laurent_coeffs(z, fz, kmin=-1, kmax=4) (or opsmath.get("cplx_laurent_coeffs"))
Laurent (and Taylor) coefficients on a uniformly sampled circle — residues included.
For `f holomorphic on an annulus around c`,
`f(z) = sum_k c_k (z - c)^k` with
`c_k = 1/(2*pi*i) ∮ f(zeta)/(zeta - c)^(k+1) dzeta`. On a circle of
radius `r sampled at n` equally spaced angles this becomes a discrete
Fourier sum, `c_k = (1/(n r^k)) sum_j f_j exp(-i k theta_j)` — the
trapezoidal rule in the angle, where it converges geometrically rather
than as `O(n^-2)` (Trefethen & Weideman 2014, "The exponentially
convergent trapezoidal rule").
`c_-1 **is the residue** at c (when c` is the only singularity
inside), `c_k for k >= 0` are the Taylor coefficients
`f^(k)(c)/k!, and a non-zero c_-m for m > 1` reveals a pole of
order `m. Measured on the unit circle with f = 1/(z - 0.5)`,
`n = 64: c_-1 = 1 and c_-2 = 0.5` to 1e-16 (machine precision).
Returns a dict: `k (int64 orders, kmin..kmax) · c` (complex128
coefficients) · `center · radius`. The centre is the sample mean,
which is exact for a uniformly sampled circle.
Orientation, and how it differs from the rest of the family: the sum
runs over the sample *set*, not the sample *order*, so this op always
returns the coefficients of the positively oriented circle — the standard
definition — whatever order the points arrive in. Feed a clockwise circle
and `c_-1 still comes back +` the residue, while
`cplx_contour_integral / (2*pi*i) on the same points returns -` it
(verified). Both are right; they answer different questions (the intrinsic
coefficient vs the integral along *this* traversal). Do not cross-check one
against the other without fixing the orientation first.
Honest limitation — aliasing: the discrete sum cannot distinguish
`c_k from c_{k+n}`, so a coefficient carries the alias sum
`sum_m c_{k+m n} r^{m n}`. That is negligible for a rapidly converging
series (the `0.5^64` term above) and ruinous near the annulus boundary.
Requesting more than `n` coefficients is refused for the same reason.
Raises `ValueError`: the samples are not a uniformly spaced circle
(unequal radii or unequal angular gaps beyond `1e-8` relative — this op
is *not* valid on an arbitrary contour, and silently pretending otherwise
would return numbers that mean nothing), `kmin > kmax, more than n`
coefficients requested, non-integer orders, and a coefficient that
overflowed (`r^-k` for a small radius and a large negative order).
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
table as input)—
complex)cplx_contour_circle · cplx_poly_eval · cplx_contour_integral · cplx_winding_number · cplx_cauchy_value · cplx_argument_principle · 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.