$\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}$
Tutorial 8 (Week 9)
Not all examples should be covered.
Definitions
- Isolated singularities
- Removable singularities
- Poles
- Essential singularities
- The Residue at $z_0$
- The Computation of Residues
- Laurent Series
- The Laurent Series of a Rational Function in Powers of $z$ and $1/z$
- Singularities at $\infty$
Problems
- 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*}
- For poles and essential singularities above derive decomposition into Laurent series and calculate the residues
Quiz 5