complex op• Data kinds: cpoints × cpoints → table
• Call: import fullseye as fs; fs.ledger.cplx_laurent_coeffs(z, fz, kmin=-1, kmax=4) (to call the implementation directly, import mathops; mathops.cplx_laurent_coeffs(z, fz, kmin=-1, kmax=4); from the registry, 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)dynsys_poincare_section · dwt_inverse · chebyshev_eval_nd
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.