Reported 2012-07-30
An effective way of computing 
in the right half-plane is
backward recurrence, beginning with a value generated from the asymptotic
expansion (5.11.3). Or we can use forward recurrence, with an
initial value obtained e.g. from (5.7.3). For the left half-plane
we can continue the backward recurrence or make use of the reflection formula
(5.5.3).
Similarly for 
, 
, and the polygamma
functions.
Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. See Schmelzer and Trefethen (2007).
For a comprehensive survey see van der Laan and Temme (1984, Chapter III). See also Borwein and Zucker (1992).
For the computation of the 
-gamma and -beta functions see Gabutti and Allasia (2008).
