A rapidly convergent series for computing $\psi (z)$ and its derivatives

Abstract:

We derive a series expansion for $\psi (z)$ in which the terms of the expansion are simple rational functions of z. From a computational viewpoint, the new series is of interest in that it converges for all z not necessarily real valued, and is particularly rapid for values of z near the origin. From a mathematical viewpoint the series is of interest in that, although $\psi (z)$ has poles at the negative integers and zero, the series is uniformly convergent in any finite interval $a < \operatorname {Re} (z) < b$.