ellipj

Jacobi elliptic functions

Syntax

• [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 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.