README.md

JacobiElliptic

JacobiElliptic is an implementation of Toshio Fukushima's & Billie C. Carlson's for calculating Elliptic Integrals and Jacobi Elliptic Functions. The default algorithms are set to the Carlson algorithms.

Features

• Type stable and preserving
• Metal.jl and CUDA.jl compatible
• Automatic Differentiable. Compatible with Zygote.jl, ForwardDiff.jl and Enzyme.jl

Documentation

Repo Status

Incomplete Elliptic Integrals

Function	Definition
F(φ, m)	$F(\phi|m)$: Incomplete elliptic integral of the first kind
E(φ, m)	$E(\phi|m)$: Incomplete elliptic integral of the second kind
Pi(n, φ, m)	$\Pi(n;\,\phi|\, m)$: Incomplete elliptic integral of the third kind
J(n, φ, m)	$J (n;\, \phi |\,m)=\frac{\Pi(n;\,\phi|\, m) - F(\phi,m)}{n}$: Associated incomplete elliptic integral of the third kind

Complete Elliptic Integrals

Function	Definition
K(m)	$K(m)$: Complete elliptic integral of the first kind
E(m)	$E(m)$: Complete elliptic integral of the second kind
Pi(n, m)	$\Pi(n|\, m)$: Complete elliptic integral of the third kind
J(n, m)	$J (n|\,m)=\frac{\Pi(n|\, m) - K(m)}{n}$: Associated incomplete elliptic integral of the third kind

Jacobi Elliptic Functions

Function	Definition
sn(u, m)	$\text{sn}(u | m) = \sin(\text{am}(u | m))$
cn(u, m)	$\text{cn}(u | m) = \cos(\text{am}(u | m))$
asn(u, m)	$\text{asn}(u | m) = \text{sn}^{-1}(u | m)$
acn(u, m)	$\text{acn}(u | m) = \text{cn}^{-1}(u | m)$