Start with and . For take integral representation (8.2.2) and
use the substitution .
The sum and the integral can be interchanged, and the sum can be evaluated via (4.6.1).
Use integration by parts.
This will result in plus two integrals with infinitely many poles.
The residue theorem (§1.10(iv)) will give us an infinite series
which can be identified via (25.11.1). For other values of and use analytic continuation.
For the Hurwitz zeta function see §25.11(i).
For other infinite series whose terms include incomplete gamma functions,
see Nemes (2017a), Reynolds and Stauffer (2021), and Prudnikov et al. (1986b, §5.2).