| course |
course_year |
question_number |
tags |
title |
year |
Complex Analysis or Complex Methods |
IB |
5 |
IB |
2020 |
Complex Analysis or Complex Methods |
|
Paper 2, Section II, B |
2020 |
For the function
$$f(z)=\frac{1}{z(z-2)}$$
find the Laurent expansions
(i) about $z=0$ in the annulus $0<|z|<2$,
(ii) about $z=0$ in the annulus $2<|z|<\infty$,
(iii) about $z=1$ in the annulus $0<|z-1|<1$.
What is the nature of the singularity of $f$, if any, at $z=0, z=\infty$ and $z=1$ ?
Using an integral of $f$, or otherwise, evaluate
$$\int_{0}^{2 \pi} \frac{2-\cos \theta}{5-4 \cos \theta} d \theta$$