# 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
    phi array_like

      amplitude of the elliptic integral

    m array_like

      parameter of the elliptic integral

  Returns
    K 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

  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.

  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.

scipy.special.ellipkm1 scipy.special.ellipe