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# scipy.special.ellipk

scipy.special.ellipk(m, out=None) = <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.

    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 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

  `elliprf`

    Completely-symmetric elliptic integral of the first 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.

  The Legendre K integral is related to Carlson’s symmetric R_F function by [2]:

  \[K(m) = R_F(0, 1-k^2, 1) .\]

  Array API Standard Support

  `ellipk` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ✅
  PyTorch | ✅ | ⛔
  JAX | ✅ | ⛔
  Dask | ✅ | n/a

  See Support for the array API standard for more information.

  References

  [1]

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

  [2]

  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