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 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])