I need help with evaluating
$f = [x(1-x)]^{\frac{1}{2}}\left|\frac{ab}{c}\frac{F(a+1,b+1;c+1;x)}{F(a,b;c;x)}\right|$
when $x \rightarrow 1$.
The result should be (according to this paper: M.S.Silver at al. Selective excitation in NMR and coherent optics through an exact solution of the Bloch-Riccati equation, Phys. Rev. A. 1985, eqs.11, 12)
$f_{x \rightarrow 1} = \left|\frac{\Gamma(a+b+1-c)\Gamma(c-a)\Gamma(c-b)}{\Gamma(a)\Gamma(b)\Gamma(c-a-b)}\right|$
Probably the parameter values of this study case can assist
$a=\frac{\phi}{2}+i\frac{\mu}{2}$
$b=-\frac{\phi}{2}+i\frac{\mu}{2}$
$c=\frac{1}{2}+i\left(\frac{\Delta\omega}{2\beta}-\frac{\mu}{2}\right)$
Maybe formula 15.4.23 can be used?