scipy.special.ellipkinc#
- scipy.special.ellipkinc(phi, m) = <ufunc 'ellipkinc'>#
Incomplete elliptic integral of the first kind
This function is defined as
\[K(\phi, m) = \int_0^{\phi} [1 - m \sin(t)^2]^{-1/2} dt\]This function is also called \(F(\phi, m)\).
- Parameters
- phiarray_like
amplitude of the elliptic integral
- marray_like
parameter of the elliptic integral
- Returns
- Kndarray
Value of the elliptic integral
See also
Notes
Wrapper for the Cephes [1] routine ellik. The computation is carried out using the arithmetic-geometric mean 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 K incomplete integral (or F integral) is related to Carlson’s symmetric R_F function [3]. Setting \(c = \csc^2\phi\),
\[F(\phi, m) = R_F(c-1, c-k^2, c) .\]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