numerics op• Data kinds: signal → signal
• Call: import fullseye as fs; fs.ledger.bessel(x, order: 'float' = 0, kind: 'str' = 'j') -> 'np.ndarray' (to call the implementation directly, import mathnumerics; mathnumerics.bessel(x, order: 'float' = 0, kind: 'str' = 'j') -> 'np.ndarray'; from the registry, opsmath.get("bessel"))
Bessel functions. kind = "j" (J of the 1st kind) / "y" (Y of the 2nd kind) / "i", "k" (modified).
Appears in vibrations of circular membranes, diffraction by circular apertures (Airy = J₁), and heat conduction in cylinders. Gate: Wronskian J_{ν+1}(x)Y_ν(x) − J_ν(x)Y_{ν+1}(x) = 2/(πx), recurrence J_{ν−1} + J_{ν+1} = (2ν/x) J_ν. Y and K only for x > 0 (they diverge at the origin).
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.
• (none yet)
signal as input)mat_solve · mat_lstsq · stat_describe · stat_histogram · stat_zscore · interp_linear · interp_cubic · interp_scattered
numerics)erf · erfc · gauss_quadrature · gauss_cubature · chebyshev_coeffs_nd · chebyshev_eval_nd · low_discrepancy · chebyshev_nodes
*Provenance: mathnumerics.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.