Jacobi elliptic functions
[SN,CN,DN] = ellipj(U,M)
[SN,CN,DN] = ellipj(U,M,tol)
The Jacobi elliptic functions are defined in terms of the integral:
Some definitions of the elliptic functions use the modulus k instead of the parameter m. They are related by
where α is the modular angle.
The Jacobi elliptic functions obey many mathematical identities; for a good sample, see .
[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).
The ellipj function is limited to the input domain 0 ≤ m ≤ 1. Map other values of M into this range using the transformations described in , equations 16.10 and 16.11. U is limited to real values.
ellipj computes the Jacobi elliptic functions using the method of the arithmetic-geometric mean of . 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:
 Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, 17.6.