Psi(x) is the digamma function,

Psi(n, x) is the nth polygamma function, which is the nth derivative of the digamma function.

You can enter the command Psi using either the 1-D or 2-D calling sequence.

If n is an integer greater than one, Psi(n) + gamma is a rational number. (gamma is Euler's constant.) For small values of n, Psi(n) computes as a sum of gamma and a rational number. To perform this computation for larger values of n, use expand.

Psi(n, x) is extended to complex n, including negative integer indices, by the formula

where $\mathrm{\zeta }$ is the Hurwitz zeta function.

$\mathrm{\Psi }\left(2\right)$

$1-\mathrm{\gamma }$

$\mathrm{\Psi }\left(1\,2\right)$

$-1+\frac{{\mathrm{\pi }}^2}{6}$

$\mathrm{\Psi }\left(3.5+4.7\mathrm{I}\right)$

$1.717883835+1.001470255\mathrm{I}$

$\mathrm{\Psi }\left(7\,-2.2+3.3\mathrm{I}\right)$

$\mathrm{-0.02713341434}+0.003825068416\mathrm{I}$

$\mathrm{\Psi }\left(-2\,1.543\right)$

$\mathrm{-0.7957394716}$

$\mathrm{\Psi }\left(1.342+\mathrm{I}\,3.5233\right)$

$\mathrm{-0.6988919005}-0.7978763419\mathrm{I}$

$\mathrm{\Psi }\left(50\right)$

$\frac{13881256687139135026631}{3099044504245996706400}-\mathrm{\gamma }$

$\mathrm{\Psi }\left(51\right)$

$\mathrm{\Psi }\left(51\right)$

$\mathrm{expand}\left(\mathrm{\Psi }\left(51\right)\right)$

$-\mathrm{\gamma }+\frac{13943237577224054960759}{3099044504245996706400}$

$\mathrm{evalf}\left(\right)$

$3.921989673$

$\mathrm{evalf}\left(\mathrm{\Psi }\left(51\right)\right)$

$3.921989673$

Espinosa, O., and Moll, V. "A Generalized Polygamma Function." Integral Transforms and Special Functions, (April 2004): 101-115.