Module bessel 

Bessel functions and radial jets with analytic argument derivatives.

Cylindrical and spherical Bessel functions of every kind (bessel), regular and singular radial jets (spherical_radial, cylindrical_radial), and their broadcasting residual. They use AMOS (the complex_bessel crate), small-argument power series, and a qualified positive-real outgoing Hankel recurrence.

Upstream: treams.special.jv, yv, hankel1, hankel2, spherical_jn and spherical_yn (scipy’s functions), and jv_d, yv_d, hankel1_d, hankel2_d, spherical_hankel1, spherical_hankel2 and their _d derivatives (_bessel.pyx).

Structs

BesselResidual

What bessel_array saves for its pullback: the orders, the arguments and the function. Only the arguments get gradients.

RadialJet

A radial function and its first two argument derivatives: a second-order jet in one complex variable, unlike the crate’s first-order jets numerics::Jet in several.

Enums

Bessel

Cylindrical Bessel solution, also used at half order for spherical functions.

Radial

Radial kind of multipole waves: regular or singular.

Constants

PARALLEL 🔒

Elementwise Bessel evaluations and pullbacks run in parallel from this many elements.

SERIES_TERMS 🔒

Terms of the small-argument power series of spherical_radial and cylindrical_radial. Below |z| = SERIES_RADIUS (0.5) each term is at most 1/(16 q^2) of its predecessor, so the last one is below 1e-40 of the first.

Functions

amos_error 🔒

bessel

Bessel value or one of its first two complex-argument derivatives.

bessel_array

Evaluate elementwise Bessel functions, broadcasting either scalar input.

bessel_raw 🔒

One AMOS evaluation of a cylindrical function, or of a spherical one at order + 1/2.

checked 🔒

jet, or an overflow error where one of its parts is not finite. Near zero the prefactor sqrt(pi / 2z) and the derivatives’ powers of 1/z overflow where AMOS does not, as for h_0 below |z| = 1e-103.

cylindrical_radial

Integer-order cylindrical J (regular) or outgoing H1 (singular), with its first two derivatives.

even_series 🔒

The even power series sum_q t_q with t_0 = leading z^power and t_q = factor(q) z^2 t_(q-1), differentiated term by term: every term reuses its predecessor’s powers.

hankel_below 🔒

Outgoing cylindrical functions [H_(m-1)(z), H_m(z)] from one AMOS sequence, with H_-1 = -H_1: the derivative u^m H_(m-1)(u) of u^m H_m(u) without the cancelling product rule.

jet_from_next_order 🔒

Derivatives of a spherical function from the next order, f' = (l/z) f - f_(l+1), and the spherical Bessel equation.

pair 🔒

The two values of a two-term sequence.

radial_jet 🔒

A spherical or cylindrical radial function of integer order at a jet argument: without derivatives a single value-only Bessel evaluation, otherwise the value and first derivative from spherical_radial or cylindrical_radial.

sequence 🔒

Cylindrical functions of orders start, start + 1, ... from one AMOS recurrence.

spherical_hankels 🔒

Spherical Hankel functions h_p(z) of the first kind for p = first..=last at a nonzero argument. Real positive arguments use the elementary outgoing seed and upward recurrence (DLMF 10.49.6, 10.51.1). Unlike regular J, H contains the dominant irregular solution, so this direction is stable below the turning point too. The bounded domain keeps every intermediate finite and agrees with AMOS to the existing 1e-13 sequence tolerance; other arguments retain one AMOS sequence.

spherical_pair 🔒

Spherical functions of orders order and order + 1. Hankel functions share one AMOS sequence, which agrees with single evaluations to 2e-14 relative. J and Y sequences differ from single evaluations by up to 1e-12, so these stay single.

spherical_prefactor 🔒

sqrt(pi / 2z), which turns cylindrical functions of order l + 1/2 into spherical functions of order l.

spherical_radial

Evaluate a spherical radial function and its first two derivatives: j_l for regular and h1_l for singular waves.

spherical_radial_sequence 🔒

spherical_radial jets of every degree 0..=order at one argument. Regular jets share power-series terms or single AMOS J evaluations. Singular jets share one spherical_hankels sequence, using its positive-real recurrence where qualified and AMOS elsewhere. Against 30-digit values at |Im z| <= 5, the AMOS sequence stays within 2e-13 relative up to order 80 (two-term sequences: 8e-14); below Im z = 0 it loses up to a digit to two-term sequences, still within 1e-13 of each jet’s size up to order 40.