# scipy.special.ellipk

scipy.special.ellipk(m) = <ufunc 'ellipk'>

  Complete elliptic integral of the first kind.

  This function is defined as

  \[K(m) = \int_0^{\pi/2} [1 - m \sin(t)^2]^{-1/2} dt\]
  Parameters
    m array_like

      The parameter of the elliptic integral.

  Returns
    K array_like

      Value of the elliptic integral.

  See also

  `ellipkm1`

    Complete elliptic integral of the first kind around m = 1

  `ellipkinc`

    Incomplete elliptic integral of the first kind

  `ellipe`

    Complete elliptic integral of the second kind

  `ellipeinc`

    Incomplete elliptic integral of the second kind

  Notes

  For more precision around point m = 1, use `ellipkm1`, which this function calls.

  The parameterization in terms of \(m\) follows that of section 17.2 in [1]. 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

    Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

scipy.special.ellipj scipy.special.ellipkm1