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
gammaincregularized lower incomplete gamma function
gammaincinvinverse of the regularized lower incomplete gamma function
gammainccinvinverse of the regularized upper incomplete gamma function
Notes
The function satisfies the relation
gammainc(a, x) + gammaincc(a, x) = 1wheregammaincis the regularized lower incomplete gamma function.The implementation largely follows that of [boost].
gammaincchas experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variableSCIPY_ARRAY_API=1and 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 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