Hermite's Formula for Hurwitz Zeta Function

Theorem

 * $\displaystyle \map \zeta {s, q} = \frac 1 {2 q^s} + \frac { q^{1 - s} } {s - 1} + 2 \int_0^\infty \frac {\map \sin {s \arctan \frac x q} } {\paren {q^2 + x^2}^{\frac 1 2 s} \paren {e^{2 \pi x} - 1} } \rd x$

where:
 * $\zeta$ is the Hurwitz zeta function
 * $\map \Re s > 1$
 * $\map \Re q > 0$.