scipy.special.ellipeinc¶

scipy.special.ellipeinc(phi, m) = <ufunc 'ellipeinc'>

Incomplete elliptic integral of the second kind

This function is defined as

$E(\phi, m) = \int_0^{\phi} [1 - m \sin(t)^2]^{1/2} dt$
Parameters
phiarray_like

amplitude of the elliptic integral.

marray_like

parameter of the elliptic integral.

Returns
Endarray

Value of the elliptic integral.

ellipkm1

Complete elliptic integral of the first kind, near m = 1

ellipk

Complete elliptic integral of the first kind

ellipkinc

Incomplete elliptic integral of the first kind

ellipe

Complete elliptic integral of the second kind

Notes

Wrapper for the Cephes [1] routine ellie.

Computation uses arithmetic-geometric means algorithm.

The parameterization in terms of $$m$$ follows that of section 17.2 in [2]. Other parameterizations in terms of the complementary parameter $$1 - m$$, modular angle $$\sin^2(\alpha) = m$$, or modulus $$k^2 = m$$ are also used, so be careful that you choose the correct parameter.

References

1

Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

2

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

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