Elliptic.jl

Elliptic integral and Jacobi elliptic special functions
Popularity
17 Stars
Updated Last
7 Months Ago
Started In
February 2013

Elliptic Special Functions for Julia

Build Status

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")

Required Packages

No packages found.

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