The below is a theorem from a book and the author says that it is a consequence of Cauchy integral formula, though I am unable to prove it. I took the rectangular contour whose vertices were $\gamma-i \beta, \alpha - i \beta, \alpha + i \beta $ and $\gamma + i \beta$ and used the Cauchy integral formula. Now I would like to let $\alpha \to \infty$ but I am unable to proceed after that. I would greatly appreciate help on this front.
