Psi(x) is the digamma function,

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

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 balanced polygamma formula of Espinosa and Moll

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

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

$1-\mathrm{\gamma }$

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

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

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

$1.717883835+1.001470255\mathrm{I}$

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

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

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

$\mathrm{-0.7957394716}$

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

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

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

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

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

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

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

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

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

$3.921989673$

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

$3.921989673$

$\mathrm{convert}\left(\mathrm{lnGAMMA}\left(x\right)\,'\mathrm{\Psi }'\right)-\mathrm{\Psi }\left(-1\,x\right)$

$\frac{\ln \left(2\mathrm{\pi }\right)}{2}$

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