Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 32C (Asymptotic Methods)

An asymptotic sequence satisfiesThe expansion
$f\sim {\sum }_{n\ge 0}{a}_n{\varphi }_n$ means that for every $N\ge 0$,For $N=0$, division by ${\varphi }_0$ givesFor $n\ge 1$, the definition with $N=n$ saysDivision by ${\varphi }_n$ gives the second coefficient formula. In particular, asymptotic-expansion coefficients are unique.

Sincewe haveThus $a_n=1$ for every $n\ge 0$.

Explicitly,For $n\ge 1$, the remainder after terms $0,\dots ,n-1$ isIts ratio to ${\varphi }_n$ tends to one, verifying every coefficient formula.

Because $e^{-x}=o(x^{-N})$ for every $N$, the exponentially small part is invisible to the power sequence ${\psi }_n=x^{-n}$. Hencewith coefficient one at ${\psi }_1$ and all other power coefficients zero.

Repeated integration by parts starts withContinuing alternates sine and cosine and multiplies by successive integers. After finitely many steps the remainder is bounded by a constant times the next inverse power, so the fixed-lower-limit sine-integral expansion isTherefore