Evaluate Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$

Question

I am at this point of integration where:

$$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$$

whereby $\beta_1$, $\beta_2$, $\sigma$ and $x$ are real and $c>0$

Cauchy's residue theorem is used and I am not sure how the integral can be simplified to apply the theorem

Answer 1

Hint: detect the poles of the integrand function with their residues, then move the integration line towards the left. Prove that, assuming $\beta_2>0$, the integral over a segment parallel to the real line vanishes as the distance from the real line goes to $+\infty$, then apply the residue theorem over a rectangular contour. See also Mellin's transform.