Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 31C / c / Solution

The substitution $u=e^{-x}$ changes the beta function toFor fixed $t>0$, its factor near zero expands asAway from zero the factor is bounded, so the extension from the finite-interval Watson lemma to this exponentially damped tail is immediate. ThereforeThe change $u\mapsto 1-u$ gives $B(s,t)=B(t,s)$, so, with $s>0$ fixed,These expansions hold with the other parameter fixed; the first does not claim uniformity when $t$ grows with $s$. The printed gamma-integral hint has inconsistent variable labels: the integration differential is $dx$ and the positive parameter is $y$.