MATLAB Function Reference |

**Syntax**

**Definition**

The Jacobi elliptic functions are defined in terms of the integral:

Some definitions of the elliptic functions use the modulus instead of the parameter . They are related by

The Jacobi elliptic functions obey many mathematical identities; for a good sample, see [1].

**Description**

```
[SN,CN,DN] = ellipj(U,M)
```

returns the Jacobi elliptic functions `SN`

, `CN`

, and `DN`

, evaluated for corresponding elements of argument `U`

and parameter `M`

. Inputs `U`

and `M`

must be the same size (or either can be scalar).

```
[SN,CN,DN] = ellipj(U,M,tol)
```

computes the Jacobi elliptic functions to accuracy `tol`

. The default is `eps`

; increase this for a less accurate but more quickly computed answer.

**Algorithm**

`ellipj`

computes the Jacobi elliptic functions using the method of the arithmetic-geometric mean [1]. It starts with the triplet of numbers:

`ellipj`

computes successive iterates with

Next, it calculates the amplitudes in radians using:

being careful to unwrap the phases correctly. The Jacobian elliptic functions are then simply:

**Limitations**

The `ellipj`

function is limited to the input domain . Map other values of `M`

into this range using the transformations described in [1], equations 16.10 and 16.11. `U`

is limited to real values.

**See Also**

**References**

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

eigs | ellipke |