Hurwitz Zeta Function - Analytic Continuation

Analytic Continuation

If Re(s) ≤ 1 the Hurwitz zeta function can be defined by the equation

where the contour C is a loop around the negative real axis. This provides an analytic continuation of .

The Hurwitz zeta function can be extended by analytic continuation to a meromorphic function defined for all complex numbers s with s ≠ 1. At s = 1 it has a simple pole with residue 1. The constant term is given by

\lim_{s\to 1} \left = \frac{-\Gamma'(q)}{\Gamma(q)} = -\psi(q)

where Γ is the Gamma function and ψ is the digamma function.

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