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Complete elliptic integrals of the first and second kind

Syntax

Definition

The complete elliptic integral of the first kind [1] is:


where F, the elliptic integral of the first kind, is:


The complete elliptic integral of the second kind,


is:


Some definitions of K and E use the modulus k instead of the parameter m. They are related by:

Description

K = ellipke(M) returns the complete elliptic integral of the first kind for the elements of M.

[K,E] = ellipke(M) returns the complete elliptic integral of the first and second kinds.

[K,E] = ellipke(M,tol) computes the Jacobian elliptic functions to accuracy tol. The default is eps; increase this for a less accurate but more quickly computed answer.

Algorithm

ellipke computes the complete elliptic integral using the method of the arithmetic-geometric mean described in [1], section 17.6. It starts with the triplet of numbers:


ellipke computes successive iterations of ai, bi, and ci with:


stopping at iteration n when cn 0, within the tolerance specified by eps. The complete elliptic integral of the first kind is then:

Limitations

ellipke is limited to the input domain .

See Also

ellipj      Jacobi elliptic functions

References

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


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