scipy.special.gammainc¶

scipy.special.gammainc(a, x) = <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

Returns

See also

gammaincc: regularized upper incomplete gamma function
gammaincinv: inverse of the regularized lower incomplete gamma function with respect to x
gammainccinv: inverse of the regularized upper incomplete gamma function with respect to x

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].

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