Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 30A / a / Solution

One form of Watson lemma is this. If $f(t)$ has the asymptotic expansion $f(t)\sim {\sum }_{n\ge 0}{a}_n{t}^{{\lambda }_n-1}$ as $t\downarrow 0$, with $0<{\lambda }_0<{\lambda }_1<\cdots$, and the Laplace integral has suitable growth control away from zero, thenas $x\to \infty$, with truncation after any fixed number of terms justified by the corresponding remainder estimate.