scipy.special.gammaincc#
- scipy.special.gammaincc(a, x, out=None) = <ufunc 'gammaincc'>#
Regularized upper incomplete gamma function.
It is defined as
\[Q(a, x) = \frac{1}{\Gamma(a)} \int_x^\infty t^{a - 1}e^{-t} dt\]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 upper incomplete gamma function
See also
gammainc
regularized lower 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
wheregammainc
is the regularized lower 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 survival function of the gamma distribution, so it starts at 1 and monotonically decreases to 0.
>>> sc.gammaincc(0.5, [0, 1, 10, 100, 1000]) array([1.00000000e+00, 1.57299207e-01, 7.74421643e-06, 2.08848758e-45, 0.00000000e+00])
It is equal to one minus the lower incomplete gamma function.
>>> a, x = 0.5, 0.4 >>> sc.gammaincc(a, x) 0.37109336952269756 >>> 1 - sc.gammainc(a, x) 0.37109336952269756