scipy.special.loggamma#

scipy.special.loggamma(z, out=None) = <ufunc 'loggamma'>#

Principal branch of the logarithm of the gamma function.

Defined to be \(\log(\Gamma(x))\) for \(x > 0\) and extended to the complex plane by analytic continuation. The function has a single branch cut on the negative real axis.

Added in version 0.18.0.

Parameters:

zarray_like: Values in the complex plane at which to compute loggamma
outndarray, optional: Output array for computed values of loggamma

Returns:

loggammascalar or ndarray: Values of loggamma at z.

See also

gammaln: logarithm of the absolute value of the gamma function
gammasgn: sign of the gamma function

Notes

It is not generally true that \(\log\Gamma(z) = \log(\Gamma(z))\), though the real parts of the functions do agree. The benefit of not defining loggamma as \(\log(\Gamma(z))\) is that the latter function has a complicated branch cut structure whereas loggamma is analytic except for on the negative real axis.

The identities

\[\begin{split}\exp(\log\Gamma(z)) &= \Gamma(z) \\ \log\Gamma(z + 1) &= \log(z) + \log\Gamma(z)\end{split}\]

make loggamma useful for working in complex logspace.

On the real line loggamma is related to gammaln via exp(loggamma(x + 0j)) = gammasgn(x)*exp(gammaln(x)), up to rounding error.

The implementation here is based on [hare1997].

Array API Standard Support

loggamma has support for Python Array API Standard compatible backends in addition to NumPy. 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

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. loggamma does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

References

[hare1997]

D.E.G. Hare, Computing the Principal Branch of log-Gamma, Journal of Algorithms, Volume 25, Issue 2, November 1997, pages 221-236.

Examples

>>> import numpy as np
>>> from scipy.special import loggamma, gamma

>>> z = 1.5 + 2j
>>> loggamma(z)
np.complex128(-1.4991963725850939+0.7332806816909994j)

Verify \(\exp(\log \Gamma(z)) = \Gamma(z)\):

>>> np.exp(loggamma(z))
np.complex128(0.165915108938991+0.14946347326641998j)
>>> gamma(z)
np.complex128(0.165915108938991+0.14946347326641998j)

Verify the recurrence \(\log \Gamma(z+1) = \log(z) + \log \Gamma(z)\):

>>> loggamma(z + 1)
np.complex128(-0.5829056407109388+1.6605758996926108j)
>>> np.log(z) + loggamma(z)
np.complex128(-0.5829056407109388+1.6605758996926117j)