# A derivation using integral Gaussian hypergeometric function

The equation I want to derive is $$\exp\left\{-2\pi\lambda\int^{\infty}_r\left(1-\frac{1}{1+zPv^{-\alpha}}\right)vdv\right\}=\exp\left\{-\pi\lambda^2\left[{}_2F_1\left(-\frac{2}{\alpha},1;1-\frac{2}{\alpha},-\frac{zP}{r^{\alpha}}\right)-1\right]\right\}$$ I was trying to use the formula below after some manipulates about changing of variables $$B(b,c-b)\,_2F_1(a,b;c;z) = \int_0^1 x^{b-1} (1-x)^{c-b-1}(1-zx)^{-a} \, dx$$ where $$B$$ is the Beta function. But I did not get the results. Any help? Thanks!