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  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 complete elliptic integral 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 , 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 .

See Also

`ellipj`

References

  Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, 17.6.

© 1994-2005 The MathWorks, Inc.