ellipj (MATLAB Function Reference)

Jacobi elliptic functions

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

Definition

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

Then

Some definitions of the elliptic functions use the modulus k instead of the parameter m. 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.