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# 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:
    x array_like

      Real argument

    out ndarray, 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`.

  Array API Standard Support

  `gammaln` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. 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

  > See Support for the array API standard for more information.

  References

  [dlmf]

  NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/5

  Examples

      >>> import numpy as np
      >>> 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])