Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 32A (Asymptotic Methods)

The functions $({\varphi }_n)$ form an asymptotic sequence as $t\to \infty$ whenfor every $n$.

The notationmeans that for every $N\ge 0$,The series need not converge for any fixed $t$.

Let $\Phi (t)=2t^4-t^2$. Its stationary points on $[0,1]$ are the endpoint $0$ and the interior point $1/2$, withThe endpoint stationary-phase contribution to ${\int }_0^1{e}^{ix\Phi (t)}dt$ isand the interior contribution isThe nonstationary endpoint contributes only $O(x^{-1})$. Taking imaginary parts gives

Write $\sinh (x\Phi )=(e^{x\Phi }-e^{-x\Phi })/2$. The global maximum of $\Phi$ is the nonstationary endpoint $t=1$, whereEndpoint Laplace's method therefore givesThe second exponential has its maximum at $t=1/2$ and is only of order $x^{-1/2}{e}^{x/8}$, exponentially smaller. Hence