|MATLAB Function Reference|
Jacobi elliptic functions
The Jacobi elliptic functions are defined in terms of the integral:
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 .
[SN,CN,DN] = ellipj(U,M)
returns the Jacobi elliptic functions
DN, evaluated for corresponding elements of argument
U and parameter
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.
ellipj computes the Jacobi elliptic functions using the method of the arithmetic-geometric mean . 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:
ellipj function is limited to the input domain . 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.
 Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, 17.6.