ellipj

Purpose

Jacobian elliptic functions.

Synopsis

[sn,cn,dn] = ellipj(u,m)

Description

[sn,cn,dn] = ellipj(u,m) returns the Jacobian elliptic functions sn, cn, and dn for the values in u evaluated at the corresponding parameters in m. Inputs u and m can contain scalars or matrices, but if they are both matrices, they must be the same size.

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

Then

Sometimes the elliptic functions are defined differently, which can lead to some confusion. The alternate definitions are usually in terms of the modulus k, which is related to m by:

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

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

Algorithm

Compute the Jacobian elliptic functions using the method of the arithmetic-geometric mean [1]. Start with the triplet of numbers:

Compute successive iterates with

Next, calculate the amplitudes in radians using

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

Limitations

ellipj is limited to the input domain 0 <= m <= 1. Other values of m can be mapped into this range using the transformations described in [1], equations 16.10 and 16.11. u is limited to real values.

See Also

ellipke

References

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

(c) Copyright 1994 by The MathWorks, Inc.