JacobiElliptic.jl

Elliptic integrals and Jacobi elliptic functions that are GPU friendly and auto differentiable
Author dominic-chang
Website Github
Popularity
7 Stars
Updated Last
4 Months Ago
Started In
December 2022

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.

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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) $\text{acn}(u | m) = \text{cn}^{-1}(u | m)$

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