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:
aarray_like

Positive parameter

xarray_like

Nonnegative argument

outndarray, 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

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
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