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:  z : array_like
Real or complex argument.
 out : ndarray, optional
Array for the computed values of
psi
.
Returns:  digamma : 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 thatpsi
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.References
[1] (1, 2, 3, 4) NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/5 [2] (1, 2) Fredrik Johansson and others. “mpmath: a Python library for arbitraryprecision floatingpoint arithmetic” (Version 0.19) http://mpmath.org/