Hermite's Formula for Hurwitz Zeta Function

Theorem

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

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

Proof
To prove this theorem, we can make use of Binet's Second Formula for Log Gamma:

Let $q$ be a complex number with a positive real part.

Then:


 * $\ds \Ln \map \Gamma q = \paren {q - \frac 1 2} \Ln q - q + \frac 1 2 \ln 2 \pi + 2 \int_0^\infty \frac {\map \arctan {x / q} } {e^{2 \pi x} - 1} \rd x$

Applying the $n$th fractional derivative with respect to $q$ on both sides, we get:

Now we can solve the fractional derivatives:

So:

Dividing both sides by the common factor, we get the initial formula:


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