Hurwitz zeta function

In mathematics, the Hurwitz zeta function, named after Adolf Hurwitz, is one of the many zeta functions. It is formally defined for complex arguments s with Re(s) > 1 and q with Re(q) > 0 by

This series is absolutely convergent for the given values of s and q and can be extended to a meromorphic function defined for all s≠1. The Riemann zeta function is ζ(s,1).

If ${\displaystyle \mathrm {Re} (s)\leq 1}$ the Hurwitz zeta function can be defined by the equation

where the contour ${\displaystyle C}$ is a loop around the negative real axis. This provides an analytic continuation of ${\displaystyle \zeta (s,q)}$.

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