Bessel function

Detailed Description

This module provides implementation for the Bessel functions and related operations.

Required header:

#include <eve/module/bessel.hpp> 

Variables
constexpr auto	eve::airy = functor<airy_t>
	elementwise_callable object computing simutaneously the airy functions values \( Ai(x)\) and \( Bi(x)\).
constexpr auto	eve::airy_ai = functor<airy_ai_t>
	elementwise_callable object computing the airy function \( Ai(x)\).
constexpr auto	eve::airy_bi = functor<airy_bi_t>
	elementwise_callable object computing the airy function \( Bi(x)\).
constexpr auto	eve::cyl_bessel_i0 = functor<cyl_bessel_i0_t>
	elementwise_callable object computing the modified Bessel function of the first kind, \( I_0(x)=\frac1{\pi}\int_{0}^{\pi}e^{x\cos\tau}\,\mathrm{d}\tau\).
constexpr auto	eve::cyl_bessel_i1 = functor<cyl_bessel_i1_t>
	elementwise_callable object computing the modified Bessel function of the first kind, \( I_1(x)=\frac1{\pi}\int_{0}^{\pi}e^{x\cos\tau}\cos\tau\,\mathrm{d}\tau\).
constexpr auto	eve::cyl_bessel_in = functor<cyl_bessel_in_t>
	elementwise_callable object computing the modified Bessel functions of the first kind, \( I_{n}(x)=\left(\frac12z\right)^n\sum_{k=0}^{\infty}{\frac{(x^2/4)^k} {k!\,\Gamma (k+n +1)}}\).
constexpr auto	eve::cyl_bessel_j0 = functor<cyl_bessel_j0_t>
	elementwise_callable object computing the Bessel function of the first kind, \( J_0(x)=\frac1{\pi }\int _{0}^{\pi}\cos(x\sin \tau) \,\mathrm {d} \tau \).
constexpr auto	eve::cyl_bessel_j1 = functor<cyl_bessel_j1_t>
	elementwise_callable object computing the Bessel function of the first kind, \( J_1(x)=\frac1{\pi }\int _{0}^{\pi}\cos(\tau-x\sin \tau )\,\mathrm {d} \tau \).
constexpr auto	eve::cyl_bessel_jn = functor<cyl_bessel_jn_t>
	elementwise_callable object computing the Bessel functions of the first kind, \( J_{n}(x)=\sum_{p=0}^{\infty}{\frac{(-1)^p}{p!\,\Gamma (p+n +1)}} {\left({x \over 2}\right)}^{2p+n }\).
constexpr auto	eve::cyl_bessel_k0 = functor<cyl_bessel_k0_t>
	elementwise_callable object computing the modified Bessel function of the second kind, \( K_0(x)=\int_{0}^{\infty}\frac{\cos(x\tau)} {\sqrt{\tau^2+1}}\,\mathrm{d}\tau\).
constexpr auto	eve::cyl_bessel_k1 = functor<cyl_bessel_k1_t>
	elementwise_callable object computing the modified Bessel function of the second kind, \( K_1(x)=\int_{0}^{\infty} e^{-x \cosh \tau} \cosh \tau\,\mathrm{d}\tau\).
constexpr auto	eve::cyl_bessel_kn = functor<cyl_bessel_kn_t>
	elementwise_callable object computing the modified Bessel function of the second kind, \( K_n(x)=\frac{\Gamma(n+1/2)(2x)^n}{\sqrt\pi} \int_{0}^{\infty}\frac{\cos\tau} {(\tau^2+x^2)^{n+1/2}}\,\mathrm{d}\tau\).
constexpr auto	eve::cyl_bessel_y0 = functor<cyl_bessel_y0_t>
	elementwise_callable object computing the Bessel function of the second kind, \( Y_0(x)=\frac2{\pi}\int_{1}^{\infty}\frac{\cos x\tau} {\sqrt{\tau^2-1}}\,\mathrm {d} \tau\).
constexpr auto	eve::cyl_bessel_y1 = functor<cyl_bessel_y1_t>
	elementwise_callable object computing the Bessel function of the second kind, \( Y_1(x)=\frac2{\pi}\int_{1}^{\infty}\frac{\cos x\tau} {(\tau^2-1)^{3/2}}\,\mathrm{d}\tau\).
constexpr auto	eve::cyl_bessel_yn = functor<cyl_bessel_yn_t>
	elementwise_callable object computing the Bessel functions of the second kind, \( Y_{n}(x)=\frac{2(z/2)^{-n}}{\sqrt\pi\, \Gamma(1/2-n)}\int _{1}^{\infty}\frac{\cos x\tau} {(\tau^2-1)^{n+1/2}}\,\mathrm {d} \tau \).
constexpr auto	eve::sph_bessel_j0 = functor<sph_bessel_j0_t>
	Computes the spherical Bessel function of the first kind and order 0, that is \( j_0(x) = \sqrt{\frac\pi{2x}}J_{1/2}(x)\) ,.
constexpr auto	eve::sph_bessel_j1 = functor<sph_bessel_j1_t>
	Computes the spherical Bessel function of the first kind of order 1,.
constexpr auto	eve::sph_bessel_jn = functor<sph_bessel_jn_t>
	Computes the spherical Bessel functions of the first kind of order n, \( j_{n}(x)= \sqrt{\frac\pi{2x}}J_{n+1/2}(x)\).
constexpr auto	eve::sph_bessel_y0 = functor<sph_bessel_y0_t>
	Computes the spherical Bessel function of the second kind of order 0, \( y_{0}(x)= \sqrt{\frac\pi{2x}}Y_{1/2}(x) \).
constexpr auto	eve::sph_bessel_y1 = functor<sph_bessel_y1_t>
	Computes the spherical Bessel function of the second kind of order 1, \( y_{1}(x)= \sqrt{\frac\pi{2x}}Y_{3/2}(x) \).
constexpr auto	eve::sph_bessel_yn = functor<sph_bessel_yn_t>
	Computes the the spherical Bessel functions of the second kind of order n, \( y_{n}(x)= \sqrt{\frac\pi{2x}}Y_{n+1/2}(x)\).