complex opcpoints × cpoints → tableimport fullseye as fs; fs.ledger.cplx_laurent_coeffs(z, fz, kmin=-1, kmax=4) (実装を直接呼ぶなら import mathops; mathops.cplx_laurent_coeffs(z, fz, kmin=-1, kmax=4)、台帳から引くなら 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.
mathops の全 op は入力を検証してから計算する(黙って通さない):
ValueError — float64 への強制変換は虚部を黙って捨てる(numpy は ComplexWarning だけ出して「もっともらしく間違った」実数を返す)。.real/.imag/abs() を明示するか、複素対応の complexops を使う。ValueError — マスクを剥がして下の生値を使う暗黙変換を拒否。埋める/落とすを明示する。ValueError(件数を明示して拒否 — 結果全体に伝播するため)。ValueError。reshape を明示する)。stat_histogram の bins は mathops.MAX_ELEMENTS(2^26 ≈ 6700 万要素)超で ValueError。py -3.11 examples/math_complex.pytable を入力に取れる)—
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. この per-op ノートは tools/opdocs.py md が自動生成(手編集しない)。
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.