caryan/elliptic.jl

Elliptic integral and Jacobi elliptic special functions

Elliptic Special Functions for Julia

This julia package provides the following:

Elliptic Integrals

K(m)	Complete elliptic integral of the first kind, K(m) = F(π/2|m)
F(phi, m)	Incomplete elliptic integral of the first kind, F(φ|m)
E(m)	Complete elliptic integral of the second kind, E(m) = E(π/2|m)
E(phi, m)	Incomplete elliptic integral of the second kind, E(φ|m)
Pi(n, phi, m)	Incomplete elliptic integral of the third kind, Π(n;φ|m)

The parameter m = k^2 = sin(α)^2 where α is the modular angle and k is the modulus.

julia> import Elliptic

julia> Elliptic.K(0.5)
1.854074677301372

Jacobi Elliptic Functions

am(u, m)	Jacobi amplitude, defined by u = F(am(u|m)|m)
sn(u, m)	Jacobi elliptic function, sn(u|m) = sin(am(u|m))
cn(u, m)	Jacobi elliptic function, cn(u|m) = cos(am(u|m))
dn(u, m)	Jacobi elliptic function, dn(u|m) = sqrt(1 - m sn(u|m)^2)
cd(u, m)	Jacobi elliptic function, cd(u|m)
sd(u, m)	Jacobi elliptic function, sd(u|m)
nd(u, m)	Jacobi elliptic function, nd(u|m) = 1/dn(u|m)
dc(u, m)	Jacobi elliptic function, dc(u|m)
nc(u, m)	Jacobi elliptic function, nc(u|m) = 1/cn(u|m)
sc(u, m)	Jacobi elliptic function, sc(u|m)
ns(u, m)	Jacobi elliptic function, ns(u|m) = 1/sn(u|m)
ds(u, m)	Jacobi elliptic function, ds(u|m)
cs(u, m)	Jacobi elliptic function, cs(u|m)

julia> import Elliptic.Jacobi

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

Matlab Compatibility

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.update()

julia> Pkg.add("Elliptic")

Definitions