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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 [1].


[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.


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 [1], equations 16.10 and 16.11. U is limited to real values.

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ellipj computes the Jacobi elliptic functions using the method of the arithmetic-geometric mean of [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:


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

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