Prove that:

$\displaystyle \int_{0}^{\infty}(\sinh{x})^{\alpha-1}P_{v}^{-\mu}(\cosh{x})\,dx = \frac{2^{-1-\mu}\Gamma\left(\frac{1}{2}\alpha+\frac{1}{2}\mu\r ight)\Gamma\left(\frac{1}{2}v-\frac{1}{2}\alpha+1\right)\Gamma\left(\frac{1}{2}-\frac{1}{2}\alpha-\frac{1}{2}v\right)}{\Gamma\left(\frac{1}{2}\mu+\f rac{1}{2}v+1\right)\Gamma\left(\frac{1}{2}+\frac{1 }{2}\mu-\frac{1}{2}v\right)\Gamma\left(1+\frac{1}{2}\mu-\frac{1}{2}\alpha\right)}$

Code:

`$\displaystyle \text{Re} \left(\alpha+\mu\right) > 0$, $\displaystyle \text{Re} \left(v-\alpha+2\right) > 0$, $\displaystyle \text{Re} (1-\alpha-v) > 0$.`