E.V.E
v2023.02.15
Bessel function

Detailed Description

Bessel functions.

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

Required header:

#include <eve/module/bessel.hpp>

Variables

constexpr callable_airy_ eve::airy = {}
 Computes the airy functions values \( Ai(x)\) and \( Bi(x)\). More...
 
constexpr callable_airy_ai_ eve::airy_ai = {}
 Computes the airy function \( Ai(x)\). More...
 
constexpr callable_airy_bi_ eve::airy_bi = {}
 Computes the airy function \( Bi(x)\). More...
 
constexpr callable_cyl_bessel_i0_ eve::cyl_bessel_i0 = {}
 Computes the modified Bessel function of the first kind, \( I_0(x)=\frac1{\pi}\int_{0}^{\pi}e^{x\cos\tau}\,\mathrm{d}\tau\). More...
 
constexpr callable_cyl_bessel_i1_ eve::cyl_bessel_i1 = {}
 Computes the modified Bessel function of the first kind, \( I_1(x)=\frac1{\pi}\int_{0}^{\pi}e^{x\cos\tau}\cos\tau\,\mathrm{d}\tau\). More...
 
constexpr callable_cyl_bessel_in_ eve::cyl_bessel_in = {}
 Computes the modified Bessel functions of the first kind, \( I_{n}(x)=\left(\frac12z\right)^n\sum_{k=0}^{\infty}{\frac{(x^2/4)^k} {k!\,\Gamma (k+n +1)}}\). More...
 
constexpr callable_cyl_bessel_j0_ eve::cyl_bessel_j0 = {}
 Computes the Bessel function of the first kind, \( J_0(x)=\frac1{\pi }\int _{0}^{\pi}\cos(x\sin \tau) \,\mathrm {d} \tau \). More...
 
constexpr callable_cyl_bessel_j1_ eve::cyl_bessel_j1 = {}
 Computes the Bessel function of the first kind, \( J_1(x)=\frac1{\pi }\int _{0}^{\pi}\cos(\tau-x\sin \tau )\,\mathrm {d} \tau \). More...
 
constexpr callable_cyl_bessel_jn_ eve::cyl_bessel_jn = {}
 Computes the Bessel functions of the first kind, \( J_{n}(x)=\sum_{p=0}^{\infty}{\frac{(-1)^p}{p!\,\Gamma (p+n +1)}} {\left({x \over 2}\right)}^{2p+n }\). More...
 
constexpr callable_cyl_bessel_k0_ eve::cyl_bessel_k0 = {}
 Computes the modified Bessel function of the second kind, \( K_0(x)=\int_{0}^{\infty}\frac{\cos(x\tau)} {\sqrt{\tau^2+1}}\,\mathrm{d}\tau\). More...
 
constexpr callable_cyl_bessel_k1_ eve::cyl_bessel_k1 = {}
 Computes the modified Bessel function of the second kind, \( K_1(x)=\int_{0}^{\infty} e^{-x \cosh \tau} \cosh \tau\,\mathrm{d}\tau\). More...
 
constexpr callable_cyl_bessel_kn_ eve::cyl_bessel_kn = {}
 Computes the modified Bessel function of the second kind, \( K_n(x)=\frac{\Gamma(n+1/2)(2x)^n}{\sqrt\pi} \int_{0}^{\infty}\frac{\cos\tau} {(\tau^2+x^2)^{n+1/2}}\,\mathrm{d}\tau\). More...
 
constexpr callable_cyl_bessel_y0_ eve::cyl_bessel_y0 = {}
 Computes the Bessel function of the second kind, \( Y_0(x)=\frac2{\pi}\int_{1}^{\infty}\frac{\cos x\tau} {\sqrt{\tau^2-1}}\,\mathrm {d} \tau\). More...
 
constexpr callable_cyl_bessel_y1_ eve::cyl_bessel_y1 = {}
 Computes the Bessel function of the second kind, \( Y_1(x)=\frac2{\pi}\int_{1}^{\infty}\frac{\cos x\tau} {(\tau^2-1)^{3/2}}\,\mathrm{d}\tau\). More...
 
constexpr callable_cyl_bessel_yn_ eve::cyl_bessel_yn = {}
 Computes the Bessel functions of the second kind, \( Y_{n}(x)=\frac{2(z/2)^{-n}}{\sqrt\pi\, \Gamma(1/2-n)}\int _{1}^{\infty}\frac{\cos x\tau} {(\tau^2-1)^{n+1/2}}\,\mathrm {d} \tau \). More...
 
constexpr callable_sph_bessel_j0_ eve::sph_bessel_j0 = {}
 Computes the spherical Bessel function of the first kind, \( j_{0}(x)= \sqrt{\frac\pi{2x}}J_{1/2}(x) \). More...
 
constexpr callable_sph_bessel_j1_ eve::sph_bessel_j1 = {}
 Computes the spherical Bessel function of the first kind, \( j_{1}(x)= \sqrt{\frac\pi{2x}}J_{3/2}(x) \). More...
 
constexpr callable_sph_bessel_jn_ eve::sph_bessel_jn = {}
 Computes the spherical Bessel functions of the first kind, \( j_{n}(x)= \sqrt{\frac\pi{2x}}J_{n+1/2}(x)\). More...
 
constexpr callable_sph_bessel_y0_ eve::sph_bessel_y0 = {}
 Computes the spherical Bessel function of the second kind, \( y_{0}(x)= \sqrt{\frac\pi{2x}}Y_{1/2}(x) \). More...
 
constexpr callable_sph_bessel_y1_ eve::sph_bessel_y1 = {}
 Computes the spherical Bessel function of the second kind, \( y_{1}(x)= \sqrt{\frac\pi{2x}}Y_{3/2}(x) \). More...
 
constexpr callable_sph_bessel_yn_ eve::sph_bessel_yn = {}
 Computes the the spherical Bessel functions of the second kind, \( y_{n}(x)= \sqrt{\frac\pi{2x}}Y_{n+1/2}(x)\). More...