# scipy.special.psi#

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

The digamma function.

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

Parameters
zarray_like

Real or complex argument.

outndarray, optional

Array for the computed values of `psi`.

Returns
digammandarray

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 . For small arguments not close to the negative real axis, the recurrence relation (5.5.2) from  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  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 , so the function should maintain full accuracy around the origin.

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)
```