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# 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 : array-like 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

  `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 `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 use `gammaln`. On the real line the two functions are related by `exp(loggamma(x)) = gammasgn(x)*exp(gammaln(x))`, though in practice rounding errors will introduce small spurious imaginary components in `exp(loggamma(x))`.

  The implementation here is based on [hare1997].

  References

  [hare1997] | (1, 2) D.E.G. Hare, Computing the Principal Branch of log-Gamma, Journal of Algorithms, Volume 25, Issue 2, November 1997, pages 221-236.