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# scipy.special.gammainc

scipy.special.gammainc(a, x, out=None) = <ufunc 'gammainc'>

  Regularized lower incomplete gamma function.

  It is defined as

  \[P(a, x) = \frac{1}{\Gamma(a)} \int_0^x t^{a - 1}e^{-t} dt\]

  for \(a > 0\) and \(x \geq 0\). See [dlmf] for details.

  Parameters:
    a array_like

      Positive parameter

    x array_like

      Nonnegative argument

    out ndarray, optional

      Optional output array for the function values

  Returns:
    scalar or ndarray

      Values of the lower incomplete gamma function

  See also

  `gammaincc`

    regularized upper incomplete gamma function

  `gammaincinv`

    inverse of the regularized lower incomplete gamma function

  `gammainccinv`

    inverse of the regularized upper incomplete gamma function

  Notes

  The function satisfies the relation `gammainc(a, x) + gammaincc(a, x) = 1` where `gammaincc` is the regularized upper incomplete gamma function.

  The implementation largely follows that of [boost].

  `gammainc` 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

  [dlmf]

  NIST Digital Library of Mathematical functions https://dlmf.nist.gov/8.2#E4

  [boost]

  Maddock et. al., “Incomplete Gamma Functions”, https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html

  Examples

      >>> import scipy.special as sc

  It is the CDF of the gamma distribution, so it starts at 0 and monotonically increases to 1.

      >>> sc.gammainc(0.5, [0, 1, 10, 100])
      array([0.        , 0.84270079, 0.99999226, 1.        ])

  It is equal to one minus the upper incomplete gamma function.

      >>> a, x = 0.5, 0.4
      >>> sc.gammainc(a, x)
      0.6289066304773024
      >>> 1 - sc.gammaincc(a, x)
      0.6289066304773024