| course |
course_year |
question_number |
tags |
title |
year |
Complex Analysis or Complex Methods |
IB |
11 |
IB |
2018 |
Complex Analysis or Complex Methods |
|
Paper 2, Section II, A |
2018 |
(a) Let $f(z)$ be a complex function. Define the Laurent series of $f(z)$ about $z=z_{0}$, and give suitable formulae in terms of integrals for calculating the coefficients of the series.
(b) Calculate, by any means, the first 3 terms in the Laurent series about $z=0$ for
$$f(z)=\frac{1}{e^{2 z}-1}$$
Indicate the range of values of $|z|$ for which your series is valid.
(c) Let
$$g(z)=\frac{1}{2 z}+\sum_{k=1}^{m} \frac{z}{z^{2}+\pi^{2} k^{2}}$$
Classify the singularities of $F(z)=f(z)-g(z)$ for $|z|<(m+1) \pi$.
(d) By considering
$$\oint_{C_{R}} \frac{F(z)}{z^{2}} d z$$
where $C_{R}={|z|=R}$ for some suitably chosen $R>0$, show that
$$\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}$$