A proof that extends Hurwitz formula into the critical strip (Q5938933)

The classical series representation NEWLINE\[NEWLINE\zeta(1-s,a)= \frac{2\Gamma(s)} {(2\pi)^s} \sum_{n=1}^\infty \frac{1} {n^s} \cos \biggl( \frac{\pi s}{2}- 2n\pi a\biggr),NEWLINE\]NEWLINE for the Hurwitz zeta-function is valid for \(\Re s> 1\) if \(0< a\leq 1\). The author modifies the usual contour integral proof to extend the validity of the formula for \(\Re s> 0\) when \(0< a< 1\). A corresponding result is obtained for the related function defined by the integral NEWLINE\[NEWLINE\eta(s,a)= \frac{1} {\Gamma(s)} \int_0^\infty \frac{x^{s-1}} {(1+e^{-x}) e^{ax}} dx.NEWLINE\]