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# scipy.special.ellipkinc

scipy.special.ellipkinc(phi, m, out=None) = <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:
    phi array_like

      amplitude of the elliptic integral

    m array_like

      parameter of the elliptic integral

    out ndarray, optional

      Optional output array for the function values

  Returns:
    K scalar 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

  `ellipe`

    Complete elliptic integral of the second kind

  `ellipeinc`

    Incomplete elliptic integral of the second kind

  `elliprf`

    Completely-symmetric elliptic integral of the first kind.

  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