An effective way of computing $\mathrm{\Gamma }\left(z\right)$ 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 $\ln \mathrm{\Gamma }\left(z\right)$, $\psi \left(z\right)$, 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 $q$-gamma and $q$-beta functions see Gabutti and Allasia (2008).