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.
See also
ellipkm1Complete elliptic integral of the first kind, near m = 1
ellipkComplete elliptic integral of the first kind
ellipkincIncomplete elliptic integral of the first kind
ellipeComplete elliptic integral of the second kind
elliprdSymmetric elliptic integral of the second kind.
elliprfCompletely-symmetric elliptic integral of the first kind.
elliprgCompletely-symmetric 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.
The Legendre E incomplete integral can be related to combinations of Carlson’s symmetric integrals R_D, R_F, and R_G in multiple ways [3]. For example, with \(c = \csc^2\phi\),
\[E(\phi, m) = R_F(c-1, c-k^2, c) - \frac{1}{3} k^2 R_D(c-1, c-k^2, c) .\]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.
- 3
NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.28 of 2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#i