| course |
course_year |
question_number |
tags |
title |
year |
Complex Analysis or Complex Methods |
IB |
10 |
IB |
2011 |
Complex Analysis or Complex Methods |
|
Paper 1, Section II, A |
2011 |
(i) Let $-1<\alpha<0$ and let
$$\begin{aligned}
&f(z)=\frac{\log (z-\alpha)}{z} \text { where }-\pi \leqslant \arg (z-\alpha)<\pi \\
&g(z)=\frac{\log z}{z} \quad \text { where }-\pi \leqslant \arg (z)<\pi
\end{aligned}$$
Here the logarithms take their principal values. Give a sketch to indicate the positions of the branch cuts implied by the definitions of $f(z)$ and $g(z)$.
(ii) Let $h(z)=f(z)-g(z)$. Explain why $h(z)$ is analytic in the annulus $1 \leqslant|z| \leqslant R$ for any $R>1$. Obtain the first three terms of the Laurent expansion for $h(z)$ around $z=0$ in this annulus and hence evaluate
$$\oint_{|z|=2} h(z) d z$$