Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 8D / Solution

The two Gamma function integrals are absolutely convergent when $\mathrm{Re}\alpha ,\mathrm{Re}\beta >0$. Consequently Fubini theorem givesMake the change of variables $u=x+y$, $t=x/(x+y)$, so $x=ut$, $y=u(1-t)$, $u>0$, $0<t<1$, and the Jacobian is $u$. Powers of positive real variables use the real logarithm, so this transformation is also valid for complex exponents. The integral factors asBoth factors converge under the same real-part assumptions. The first is $\Gamma (\alpha +\beta )$, proving the Beta function identity