Integration with Beta Function $\beta$

Question

Given that:

$$\int_{c\ -\ j\infty}^{c\ +\ j\infty}\left({\sigma\,x^{-1}}\right)^u\beta\left(u,a\right)du=\left(1-{x \over \sigma}\right)^{a-1}$$

whereby $\sigma>0$, $a>0$ and $x$ is a real number which could be both positive or negative

I need help proving this statement using Cauchy's residue theorem

Answer 1

$\beta(u,a)$ is a Mellin transform of $(1−x)^{a−1}$ for $x∈(0,1)$ and zero for $x≥1$. $$β(u,a)=\int_{0}^{1}x^{u−1}(1−x)^{a−1}dx$$ Mellin inversion: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty}\left(x \over \sigma\right)^{−u}β\left(u,a\right)du=\left(1−{x\over \sigma}\right)^{a−1}$$