scipy.special.ellipeinc#
- scipy.special.ellipeinc(phi, m, out=None) = <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. 
- outndarray, optional
- Optional output array for the function values 
 
- Returns:
- Escalar or ndarray
- Value of the elliptic integral. 
 
 - See also - 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 
- elliprd
- Symmetric elliptic integral of the second kind. 
- elliprf
- Completely-symmetric elliptic integral of the first kind. 
- elliprg
- Completely-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