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.
New in version 0.18.0.
Parameters:  z : arraylike
Values in the complex plain at which to compute
loggamma
 out : ndarray, optional
Output array for computed values of
loggamma
Returns:  loggamma : ndarray
Values of
loggamma
at z.
See also
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 whereasloggamma
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
loggama
useful for working in complex logspace. However,loggamma
necessarily returns complex outputs for real inputs, so if you want to work only with real numbers usegammaln
. On the real line the two functions are related byexp(loggamma(x)) = gammasgn(x)*exp(gammaln(x))
, though in practice rounding errors will introduce small spurious imaginary components inexp(loggamma(x))
.The implementation here is based on [hare1997].
References
[hare1997] (1, 2) D.E.G. Hare, Computing the Principal Branch of logGamma, Journal of Algorithms, Volume 25, Issue 2, November 1997, pages 221236.