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

Tutorial 8 (Week 9)

Not all examples should be covered.


  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 $\infty$


  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