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
 zarraylike
Values in the complex plain at which to compute
loggamma
 outndarray, optional
Output array for computed values of
loggamma
 Returns
 loggammandarray
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
loggamma
useful for working in complex logspace.On the real line
loggamma
is related togammaln
viaexp(loggamma(x + 0j)) = gammasgn(x)*exp(gammaln(x))
, up to rounding error.The implementation here is based on [hare1997].
References
 hare1997
D.E.G. Hare, Computing the Principal Branch of logGamma, Journal of Algorithms, Volume 25, Issue 2, November 1997, pages 221236.