Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 30A / b / Solution

PutThe two endpoints $t=1$ and $t=-1$ both contribute. Near $t=1$, set $u=1-t$. ThenExpanding the last amplitude in powers of $u$ and applying the Complex Watson lemma after rotating onto a decaying endpoint ray gives one exponential series. Repeating at $t=-1$ gives the other. After multiplication by the prefactor in the integral representation, the two endpoint contributions combine aswhereSeparating the even and odd powers gives the large-argument asymptotic expansion of the Bessel function of the first kind:Its first terms areFor fixed $\nu$, the expansion is uniform in every closed sectorwith $\delta >0$. The exclusion of the negative real axis fixes the branch of $z^{1/2}$ and stays away from its Stokes boundary.