# ellipke

## Purpose

Complete elliptic integrals of the first and second kind.

## Synopsis

    K = ellipke(m)
    [K,E] = ellipke(m)

## Description

`[K,E] = ellipke(m)` returns the complete elliptic integral of the first and second kinds for the elements of m. The complete elliptic integral of the first kind, `K(m) = F(pi/2|m)`, is defined by

The complete elliptic integral of the second kind, `E(m) = E(K(m)) =
E(pi/2|m)`, is defined by

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

The accuracy of the result is `eps`; the value of `eps` can be changed for a less accurate, but more quickly computed answer.

## Algorithm

The complete elliptic integral is computed using the method of the arithmetic-geometric mean described in [1], section 17.6. Start with the triplet of numbers:

Compute successive iterations of ai, bi, and ci with

stopping at iteration n when cn \xc5 0, within the tolerance specified by `eps`. The complete elliptic integral of the first kind is then

## Limitations

`ellipk`e is limited to the input domain 0 <= m<= 1.

## See Also

    ellipj

## References

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

(c) Copyright 1994 by The MathWorks, Inc.