MATLAB Function Reference |

Complete elliptic integrals of the first and second kind

**Syntax**

**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

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 .

**See Also**

**References**

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

ellipj | ellipsoid |