The first paragraph of Section 2.9. of Titchmarsh's book The Theory of the Riemann Zeta-Function is a follows :
Even though the Kloosterman's paper is written in German (and I don't speak German), however I tried to find the paper to get a hint if I can prove the identity but it is very much old and not accessible to me. However, I could prove the real case by other means that I couldn't extend to complex case. In addition, the other claim after the identity is not clear as well. My questions are:
1- How Eq. 2.9.1 is derived?
2- How the integrand is $O(|z|^{-\sigma -1})$ is derived? In section 12.16 of Whittaker-Watson's book A Course of Modern Analysis, expansion of $\dfrac{\Gamma'(z+1)}{\Gamma(z+1)}$ is derived but lacks a $\log(z)$ term or I couldn't find a cancelable expansion of $\log(z)$ anywhere.