Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 8B / i / Solution

For the Euler integral to converge directly, assume $\mathrm{Re}b>0$ and $\mathrm{Re}(c-b)>0$. At zero it is the beta function $B(b,c-b)=\Gamma (b)\Gamma (c-b)/\Gamma (c)$. The normalization is thereforeOutside these initial parameter conditions the hypergeometric function is defined by analytic continuation wherever the parameters permit it.