Bessel functions

Bessels.besselj — Function

besselj(ν::Real, x::Real)
besselj(ν::AbstractRange, x::Real)

Returns the Bessel function, $J_ν(x)$, of the first kind and order ν.

\[J_{\nu}(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m!\Gamma(m+\nu+1)}(\frac{x}{2})^{2m+\nu}\]

Routine supports single and double precision (e.g., Float32 or Float64) real arguments.

For ν isa AbstractRange, returns a vector of type float(x) using recurrence to compute $J_ν(x)$ at many orders as long as the conditions ν[1] >= 0 and step(ν) == 1 are met. Consider the in-place version besselj! to avoid allocation.

Examples

julia> besselj(2, 1.5)
0.2320876721442147

julia> besselj(3.2, 2.5)
0.17888479721039752

julia> besselj(1:3, 2.5)
3-element Vector{Float64}:
 0.4970941024642743
 0.44605905843961735
 0.2166003910391135

External links: DLMF, Wikipedia

Bessels.besselj! — Function

Bessels.besselj!(out::DenseVector{T}, ν::AbstractRange, x::T)

Computes the Bessel function, $j_ν(x)$, of the first kind at many orders ν in-place using recurrence. The conditions ν[1] >= 0 and step(ν) == 1 must be met.

Examples

julia> nu = 1:3; x = 1.5; out = zeros(typeof(x), length(nu));

julia> Bessels.besselj!(out, nu, x)
3-element Vector{Float64}:
 0.5579365079100995
 0.2320876721442147
 0.06096395114113963

Modified Bessel functions

Bessels.besseli — Function

besseli(ν::Real, x::Real)
besseli(ν::AbstractRange, x::Real)

Returns the modified Bessel function, $I_ν(x)$, of the first kind and order ν.

\[I_{\nu}(x) = \sum_{m=0}^{\infty} \frac{1}{m!\Gamma(m+\nu+1)}(\frac{x}{2})^{2m+\nu}\]

Routine supports single and double precision (e.g., Float32 or Float64) real arguments.

For ν isa AbstractRange, returns a vector of type float(x) using recurrence to compute $I_ν(x)$ at many orders as long as the conditions ν[1] >= 0 and step(ν) == 1 are met. Consider the in-place version besseli! to avoid allocation.

Examples

julia> besseli(2, 1.5)
0.3378346183356807

julia> besseli(3.2, 2.5)
0.3772632469352918

julia> besseli(1:3, 2.5)
3-element Vector{Float64}:
 2.5167162452886984
 1.2764661478191641
 0.47437040877803555

External links: DLMF, Wikipedia

Bessels.besseli! — Function

Bessels.besseli!(out::DenseVector{T}, ν::AbstractRange, x::T)

Computes the modified Bessel function, $I_ν(x)$, of the first kind at many orders ν in-place using recurrence. The conditions ν[1] >= 0 and step(ν) == 1 must be met.

Examples

julia> nu = 1:3; x = 1.5; out = zeros(typeof(x), length(nu));

julia> Bessels.besseli!(out, nu, x)
3-element Vector{Float64}:
 0.9816664285779074
 0.3378346183356807
 0.0807741130160923

Hankel functions

Bessels.besselh — Function

besselh(nu, [k=1,] x)

Returns the Bessel function of the third kind of order nu (the Hankel function).

\[H^{(1)}_{\nu}(x) = J_{\nu}(x) + i Y_{\nu}(x) H^{(2)}_{\nu}(x) = J_{\nu}(x) - i Y_{\nu}(x)\]

k must be 1 or 2, selecting hankelh1 or hankelh2, respectively.

Examples

julia> besselh(1.2, 1, 9.2)
0.2513215427211038 + 0.08073652619125624im

Bessels.hankelh1 — Function

hankelh1(nu, x)

Bessel function of the third kind of order nu, $H^{(1)}_\nu(x)$.

Bessels.hankelh2 — Function

hankelh2(nu, x)

Bessel function of the third kind of order nu, $H^{(2)}_\nu(x)$.

Spherical Bessel functions

Bessels.sphericalbesselj — Function

sphericalbesselj(ν, x)

Spherical bessel function of the first kind of order ν, $j_ν(x)$. This is the non-singular solution to the radial part of the Helmholz equation in spherical coordinates.

\[j_{\nu}(x) = \sqrt(\frac{\pi}{2x}) J_{\nu + 1/2}(x)\]

where $J_{\nu}$ is the Bessel function of the first kind. Routine supports single and double precision (e.g., Float32, Float64) for real arguments.

Examples

julia> sphericalbesselj(1, 1.2)
0.34528456985779027

julia> sphericalbesselj(1.5, 1.2)
0.18231344932382995

External links: DLMF, Wikipedia

Airy functions

Bessels.airyai — Function

airyai(z)

Returns the Airy function of the first kind, $\operatorname{Ai}(z)$, which is the solution to the Airy differential equation $f''(z) - z f(z) = 0$.

\[\operatorname{Ai}(z) = \frac{\sqrt{3}}{2 \pi} \int_{0}^^{\infty} \exp{-\frac{t^3}{3} - \frac{z^3}{3t^3}} dt\]

Routine supports single and double precision (e.g., Float32, Float64, ComplexF64) for real and complex arguments.

Examples

julia> airyai(1.2)
0.10612576226331255

julia> airyai(1.2 + 1.4im)
-0.03254458873613304 - 0.14708163733976673im

External links: DLMF, Wikipedia

Gamma functions

Bessels.gamma — Function

gamma(x::T) where T <: Union{Float16, Float32, Float64}

Returns the gamma function, $Γ(x)$, for real x.

\[\Gamma(x) = \int_x^\infty e^{-t} t^{x-1} dt \,\]

External links: DLMF, Wikipedia