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5 Gamma FunctionComputation

§5.21 Methods of Computation

An effective way of computing Γ(z) 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 lnΓ(z), ψ(z), 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).