Elliptic.jl

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Elliptic integral and Jacobi elliptic special functions

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Elliptic Special Functions for Julia
====================================

[![Build Status](https://travis-ci.org/nolta/Elliptic.jl.svg?branch=master)](https://travis-ci.org/nolta/Elliptic.jl)

This julia package implements
[elliptic integrals](https://dlmf.nist.gov/19.2) and
(in the `Jacobi` sub-module) the
[Jacobi elliptic functions](https://dlmf.nist.gov/22.2).

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](https://en.wikipedia.org/wiki/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)` <br> `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}](http://mathurl.com/av9eou5.png)

![E(\phi|m) = \int_0^\phi d\theta (1 - m\sin^2\theta)^{1/2}](http://mathurl.com/al2zsok.png)

![\Pi(n;\varphi|m) = \int_0^\varphi d\theta\, (1-n\sin^2\theta)^{-1}(1 - m\sin^2\theta)^{-1/2}](http://mathurl.com/bzsx5tw.png)

```jlcon
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)|

```jlcon
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.

```jlcon
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
------------

```jlcon
julia> Pkg.add("Elliptic")
```

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