Paper 2, Section II, A

(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

$\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}$

*Typos? Please submit corrections to this page on GitHub.*