Hermite's Formula for Hurwitz Zeta Function

Theorem

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

where:
 * $\zeta$ is the Hurwitz zeta function
 * $\operatorname{Re} \left({s}\right) > 1$
 * $\operatorname{Re} \left({q}\right) > 0$.