$\renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}}$

Tutorial 8 (Week 9)

Not all examples should be covered.

Definitions

1. Isolated singularities
2. Removable singularities
3. Poles
4. Essential singularities
5. The Residue at $z_0$
6. The Computation of Residues
7. Laurent Series
8. The Laurent Series of a Rational Function in Powers of $z$ and $1/z$
9. Singularities at $\infty$

Problems

1. Classify singularities (removable, poles, essential; others: not isolated, branching); for poles indicate their orders \begin{align*} &\frac{\tan(z)}{z} &&&&\frac{\cosh(z)+\cos(z)-2}{z^6} &&\text{at $z\ne \infty$}\\ &\sin (z) &&&&\tan (z) &&\text{at $z=\infty$}\\[6pt] &\sqrt{z} &&\sqrt[3]{z} &&\log (z) &&\text{at $z=0,\infty$} \end{align*}
2. For poles and essential singularities above derive decomposition into Laurent series and calculate the residues

Quiz 5