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# scipy.special.digamma

scipy.special.digamma(z, out=None) = <ufunc 'psi'>

  The digamma function.

  The logarithmic derivative of the gamma function evaluated at `z`.

  Parameters:
    z array_like

      Real or complex argument.

    out ndarray, optional

      Array for the computed values of `psi`.

  Returns:
    digamma scalar or ndarray

      Computed values of `psi`.

  Notes

  For large values not close to the negative real axis, `psi` is computed using the asymptotic series (5.11.2) from [1]. For small arguments not close to the negative real axis, the recurrence relation (5.5.2) from [1] is used until the argument is large enough to use the asymptotic series. For values close to the negative real axis, the reflection formula (5.5.4) from [1] is used first. Note that `psi` has a family of zeros on the negative real axis which occur between the poles at nonpositive integers. Around the zeros the reflection formula suffers from cancellation and the implementation loses precision. The sole positive zero and the first negative zero, however, are handled separately by precomputing series expansions using [2], so the function should maintain full accuracy around the origin.

  Array API Standard Support

  `psi` 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

  [1] (1,2,3)

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

  [2]

  Fredrik Johansson and others. “mpmath: a Python library for arbitrary-precision floating-point arithmetic” (Version 0.19) http://mpmath.org/

  Examples

      >>> from scipy.special import psi
      >>> z = 3 + 4j
      >>> psi(z)
      (1.55035981733341+1.0105022091860445j)

  Verify psi(z) = psi(z + 1) - 1/z:

      >>> psi(z + 1) - 1/z
      (1.55035981733341+1.0105022091860445j)