MATLAB Function Reference
ellipke

Complete elliptic integrals of the first and second kind

Syntax

• ```K = ellipke(M)
[K,E] = ellipke(M)
[K,E] = ellipke(M,tol)
```

Definition

The complete elliptic integral of the first kind [1] is

where , the elliptic integral of the first kind, is

The complete elliptic integral of the second kind

is

Some definitions of `K` and `E` use the modulus instead of the parameter . They are related by

Description

```K = ellipke(M) ``` returns the complete elliptic integral of the first kind for the elements of `M`.

```[K,E] = ellipke(M) ``` returns the complete elliptic integral of the first and second kinds.

```[K,E] = ellipke(M,tol) ``` computes the Jacobian elliptic functions to accuracy `tol`. The default is `eps`; increase this for a less accurate but more quickly computed answer.

Algorithm

`ellipke` computes the complete elliptic integral using the method of the arithmetic-geometric mean described in [1], section 17.6. It starts with the triplet of numbers

`ellipke` computes successive iterations of , , and with

stopping at iteration when , within the tolerance specified by `eps`. The complete elliptic integral of the first kind is then

Limitations

`ellipke` is limited to the input domain .

`ellipj`