A recent question [On a certain integral representation for Dirichlet L-functions](https://mathoverflow.net/questions/303971/on-a-certain-integral-representation-for-dirichlet-l-functions) referenced an integral representation of $\zeta(s)$ due to Jensen that was new to me:
$$
(s-1)\zeta(s)=\frac{\pi}{2(s-1)}\int\_{-\infty}^\infty\frac{(1/2+it)^{1-s}}{\cosh^2(\pi t)}\, dt.
$$
Looking for analogs for the Hurwitz zeta function, I found some interesting integral representations due to Hermite in Whittaker and Watson, based on Plana's Theorem. (Also found in Wikipedia under the Abel-Plana formula). Exercises there include still more representations of $\zeta(s)$ due to Jensen, published in *L'Intermediaire des Mathematicians*, 1895. In that publication I found the result above; towards the proof Jensen only says "proven with the aid of Cauchy's Theorem."


Internet searching eventually led me to (23) on p. 92 of *Series Associated with Zeta and Related Functions*, by Srivastava and Choi, Kluwer, 2001: for $a>1/2$,
$$
\zeta(s,a)=\frac{\pi}{2(s-1)}\int\_{-\infty}^\infty\frac{(a-1/2+it)^{1-s}}{\cosh^2(\pi t)}\, dt.
$$
No proof is given. For $a=1$ this reduces to Jensen.


Can anyone supply details of the proof?




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A side note, the journal *L'Intermediaire des Mathematicians* was new to me. It seems to be a 19th century print analog of MathOverflow, consisting of questions posed by researchers, and answers to questions from previous issues. Jensen was writing in reference to a question of Cesaro.



On a certain integral representation for Hurwitz zeta functions
All these are special cases of the Abel--Plana formula: under simple regularity
and growth condition which are easily given, for any $a$ with $\Re(a)>-1/2$ we have
$$\sum\_{n\ge 0}f'(n+a)=-\dfrac{\pi}{2}\int\_{-\infty}^\infty \dfrac{f(a-1/2+it)}{\cosh^2(\pi t)}\,dt$$
plus some correction terms due to the possible poles of $f'$ in the right
half-plane, if any.


