The function was introduced in Hurwitz (1882) and defined by the series expansion
has a meromorphic continuation in the -plane, its only singularity in being a simple pole at with residue 1. As a function of , with () fixed, is analytic in the half-plane . The Riemann zeta function is a special case:
For most purposes it suffices to restrict because of the following straightforward consequences of (25.11.1):
Most references treat real with .
For see §24.2(iii).
Throughout this subsection .
where the integration contour is a loop around the negative real axis as described for (25.5.20).
See also Srivastava and Choi (2001).
where is Catalan’s constant:
As with fixed,
As with fixed, ,
uniformly with respect to bounded nonnegative values of .
As in the sector , with and fixed, we have the asymptotic expansion
Similarly, as in the sector ,
For the more general case , , see Elizalde (1986).