 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 4

Near the endpoint $t=0$,
$$
\frac1{1+t}\sim\sum_{n=0}^\infty(-1)^nt^n.
$$
The <Watson lemma> integrates this local expansion term by term, and
$$
\int_0^\infty e^{-xt}t^n\,dt
=\frac{\Gamma(n+1)}{x^{n+1}}
=\frac{n!}{x^{n+1}}.
$$
Therefore the <reciprocal-amplitude Laplace integral> has expansion
$$
\boxed{
U(x)\sim\sum_{n=0}^\infty
(-1)^nn!x^{-n-1}},
\qquad x\to+\infty,
$$
so
$$
\boxed{a_n=(-1)^nn!}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 31D / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 31D / a / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 31D / a / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)