Elliptic Special Functions for Julia

This julia package implements elliptic integrals and (in the Jacobi sub-module) the Jacobi elliptic functions.

Elliptic Integrals

Function	Definition
`F(phi, m)`	Incomplete elliptic integral of the first kind, F(φ \| m)
`K(m)`	Complete elliptic integral of the first kind, the quarter period, F(π/2 \| m)
`E(phi, m)`	Incomplete elliptic integral of the second kind, E(φ \| m)
`E(m)`	Complete elliptic integral of the second kind, E(π/2 \| m)
`Pi(n, phi, m)Π(n, phi, m)`	Incomplete elliptic integral of the third kind, Π(n; φ \| m)

Where the parameter m = k^2 = sin(α)^2, α is the modular angle, k is the modulus, and

$F(\phi|m) = \int_0^\phi d\theta (1 - m\sin^2\theta)^{-1/2}$

$E(\phi|m) = \int_0^\phi d\theta (1 - m\sin^2\theta)^{1/2}$

$\Pi(n;\varphi|m) = \int_0^\varphi d\theta, (1-n\sin^2\theta)^{-1}(1 - m\sin^2\theta)^{-1/2}$

julia> import Elliptic

julia> Elliptic.K(0.5)
1.854074677301372

Jacobi Elliptic Functions

Function	Definition
`am(u, m)`	Solution to u = F(am(u \| m) \| m)
`sn(u, m)`	sn(u \| m) = sin(am(u \| m))
`cn(u, m)`	cn(u \| m) = cos(am(u \| m))
`dn(u, m)`	dn(u \| m) = sqrt(1 - m sn(u \| m)^2)
`sd(u, m)`	sd(u \| m) = sn(u \| m) / dn(u \| m)
`cd(u, m)`	cd(u \| m) = cn(u \| m) / dn(u \| m)
`nd(u, m)`	nd(u \| m) = 1 / dn(u \| m)
`dc(u, m)`	dc(u \| m) = 1 / cd(u \| n)
`nc(u, m)`	nc(u \| m) = 1 / cn(u \| m)
`sc(u, m)`	sc(u \| m) = sn(u \| m) / cn(u \| m)
`ns(u, m)`	ns(u \| m) = 1 / sn(u \| m)
`ds(u, m)`	ds(u \| m) = 1 / sd(u \| m)
`cs(u, m)`	cs(u \| m) = 1 / sc(u \| m)

julia> import Elliptic.Jacobi

julia> Jacobi.sn(2, 9)
-0.15028246569211734

Matlab Compatibility

Function	Definition
`ellipj(u, m)`	returns `(sn(u,m), cn(u,m), dn(u,m))`
`ellipke(m)`	returns `(K(m), E(m))`

For convenience, the matlab compatible ellipj and ellipke routines are also provided. ellipj(u,m) is equivalent to sn(u,m), cn(u,m), dn(u,m), but faster if you want all three. Likewise, ellipke(m) is equivalent to K(m), E(m), but faster if you want both.

julia> import Elliptic

julia> k,e = Elliptic.ellipke(0.5)
(1.854074677301372,1.3506438810476757)

julia> sn,cn,dn = Elliptic.ellipj(0.672, 0.36)
(0.6095196917919022,0.792770928653356,0.9307281387786907)

Installation

julia> Pkg.add("Elliptic")