Module legendre 

Associated Legendre functions of integer degree and the vector-wave angular functions pi and tau, with analytic argument derivatives. Real-degree Legendre values come from the Ferrers functions in ferrers.

Upstream: treams.special.lpmv, pi_fun and tau_fun (_waves.pyx).

Structs

AngularResidual

What angular_array saves for its pullback: the degrees, orders, arguments and function. The pullback recomputes the local derivatives.

Enums

Angular

Associated Legendre polynomial or one of the two vector-wave angular functions.

Constants

PARALLEL 🔒

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

Functions

angular_array

Broadcast angular values and a complex-argument residual; labels stay fixed.

angular_jet 🔒

The angular function kind of degree l and order m at z, with its derivative for N = 1, after checking the labels and the argument.

angular_jets 🔒

The angular functions [pi, tau] from the polar cosine and sine, with the sine factor kept explicit at the poles.

angular_legendre 🔒

F_l^m(z) sin(theta)^power: the Legendre function for power = |m|, and pi / m for power = |m| - 1.

angular_value

Angular function at a cosine argument; real Legendre values also accept real degrees.

legendre_factor 🔒

Associated Legendre polynomial F_l^m after factoring out sin(theta)^|m|.

legendre_factor_pair 🔒

F_l^m and F_(l-1)^m of legendre_factors, with F_(l-1)^m = 0 for l = |m| and both zero for |m| > l.

legendre_factors 🔒

Associated Legendre functions F_l^m after factoring out sin(theta)^|m|, visited for every l in |m|..=max in order from one run of the degree recurrence. Keeping the factor explicit avoids 0/0 and cancellation in angular functions near the poles.

lpmv_real

Associated Legendre function P_degree^order(x) of a real argument, as a real value.

polar_trig 🔒

[cos(theta), sin(theta)] with their derivatives. The sine takes the principal branch (nonnegative real part) and is exactly zero at the poles, where theta is a floating-point multiple of PI.

polarized_angular 🔒

Angular factor coupling the vector spherical wave (l, m) of polarization mode_pol to a plane wave of polarization pol whose direction has the polar cosine and sine given: tau -+ pi for negative (0) or positive (1) helicity; in parity polarization tau for the mode’s own polarization and pi for the other. The polarization module states this rule for the vector waves themselves.

sine_power 🔒

sin(theta)^power = (1 - z^2)^(power / 2) at the cosine z.

tau_factor 🔒

tau_l^m / sin(theta)^(|m|-1) for m != 0 from one Legendre recurrence, by DLMF 14.10.5: sin(theta) tau = l cos(theta) P_l^m - (l + m) P_(l-1)^m.