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
- aarray_like
Positive parameter
- xarray_like
Nonnegative argument
- 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 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
wheregammaincc
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