Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 8B (Further Complex Methods)

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.

For $|z|<1$, differentiation under the convergent Euler integral giveswhere $(a{)}_n=a(a+1)\cdots (a+n-1)$ and $(a{)}_0=1$ are Pochhammer symbols. Thus its Taylor series iswhich is symmetric in $a,b$. In the common region of parameter convergence for the two Euler integrals, this proves equality near zero. The identity theorem extends it throughout their common principal slit domain, and parameter continuation gives the usual identity wherever both sides are defined: