scipy.special.gammaln#

scipy.special.gammaln(x, out=None) = <ufunc 'gammaln'>#

Logarithm of the absolute value of the gamma function.

Defined as

\[\ln(\lvert\Gamma(x)\rvert)\]

where \(\Gamma\) is the gamma function. For more details on the gamma function, see [dlmf].

Parameters:

xarray_like: Real argument
outndarray, optional: Optional output array for the function results

Returns:

scalar or ndarray: Values of the log of the absolute value of gamma

See also

gammasgn: sign of the gamma function
loggamma: principal branch of the logarithm of the gamma function

Notes

It is the same function as the Python standard library function math.lgamma.

When used in conjunction with gammasgn, this function is useful for working in logspace on the real axis without having to deal with complex numbers via the relation exp(gammaln(x)) = gammasgn(x) * gamma(x).

For complex-valued log-gamma, use loggamma instead of gammaln.

References

[dlmf]

NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/5

Examples

>>> import numpy as np
>>> import scipy.special as sc

It has two positive zeros.

>>> sc.gammaln([1, 2])
array([0., 0.])

It has poles at nonpositive integers.

>>> sc.gammaln([0, -1, -2, -3, -4])
array([inf, inf, inf, inf, inf])

It asymptotically approaches x * log(x) (Stirling’s formula).

>>> x = np.array([1e10, 1e20, 1e40, 1e80])
>>> sc.gammaln(x)
array([2.20258509e+11, 4.50517019e+21, 9.11034037e+41, 1.83206807e+82])
>>> x * np.log(x)
array([2.30258509e+11, 4.60517019e+21, 9.21034037e+41, 1.84206807e+82])