I have asked related questions on MSE and received no answer, so: I am looking for proofs of the integrals presented in Erdélyi's *Table of Integral Transforms* vol. i-ii, specifically proofs of the various gamma integrals presented on page 297-299 of volume 2, for instance:


$$\int\_{\mathbb{R}} \frac{dx}{\Gamma(\alpha + x)\Gamma(\beta - x)}= \frac{2^{\alpha + \beta -2}}{\Gamma(\alpha + \beta - 1)},~\Re(\alpha + \beta)>1.$$


That the proofs for the identities presented in the work were reviewed closely is mentioned in the preface, though I cannot find any separate references containing the proofs.


[Link to e-PDFs of the work in question](https://authors.library.caltech.edu/43489/).



Reference request: proofs of integrals presented in Erdélyi's *Table of Integral Transforms*
Titchmarsh’s *Fourier integrals* ([1937](//zbmath.org/?q=an:0017.40404), [7.6.4](//archive.org/details/IntroductionToTheTheoryOfFourierIntegrals/page/n196)) has proof and attribution to Ramanujan.


