I am trying to derivethe $N$ defined via the following integral:
$$\displaystyle \frac{4\pi^2}{\omega N^2}= \int_0^L \frac{dz}{z} \left(1-\frac{z^2}{L^2}\right)^{-1} |V_k(z)|^2 $$
where $$\displaystyle V_k(z) = \left(\frac{z}{L}\right)^\Delta \left(1 - \frac{z^2}{L^2} \right)^{-iL\omega/2} {}_2 F_1\hspace{-4pt}\left[\frac{\Delta-iL\omega +iL k}{2}, \frac{\Delta-iL\omega-iL k}{2} ; \Delta ; \frac{z^2}{L^2} \right] . $$ Here $\Delta,\omega,L,k$ are all real. $\Delta$ and $L$ are positive.
The $V_k(z)$ are real and satisfy $V_{\omega,k}(z) = V_{-\omega,k}(z) = V_{\omega,-k}(z) = V_{-\omega,-k}(z)$.
The correct answer is apparently:
$$ N= \frac{1}{\sqrt{2 L |\omega|}} \frac{ \Gamma\left(\frac{\Delta - iL (\omega + k)}{2}\right) \Gamma\left(\frac{\Delta - iL (\omega - k)}{2}\right) } {\Gamma(\Delta)\Gamma(iL \omega)}$$
I want to derive this answer from the above equations. I think this should come down to some identities involving hypergeometric functions. Any help would be welcome.