28  Singularities and contour integrals

Laurent series classify isolated singularities and determine contour integrals around poles.

Exercises

Singularities

Exercise 28.1 Consider the function \(f(z) = \frac{1}{z(z-1)}\). How many poles does \(f\) have, and where are they? What are the order of the poles? Can you write down the Laurent series of \(f\) around \(z=0\)?

Exercise 28.2 Does the function \(f(z) = 1/\sin(z)\) have a pole? Where? What order?

Exercise 28.3 Find the singularities of \(f(z) = e^{-1/(z-1)^2}\).

Line integrals

Let

\[ f(z)=\frac{1}{z^3-(2+\mathrm{i})z^2+(1+2\mathrm{i})z-\mathrm{i}}. \]

Exercise 28.4 Let \(\Gamma_\epsilon(w)\) be the simple closed curve defined by the counter-clockwise border of the \(\epsilon\)-ball \(B_\epsilon(w)\). Write down a parameterization (a path) for this curve.

Exercise 28.5 Identify the poles \(P = \{w_1, w_2, \cdots \}\) of \(f(z)\) and their order. Hint: A zero of the denominator is \(z = \mathrm{i}\).

Exercise 28.6 What is the value of \[\oint_{\Gamma_\epsilon(0)} f(z)\, \mathrm{d}z\] for \(\epsilon = 1/2\)?

Exercise 28.7 Find the value of the line integrals \[\oint_{\Gamma_\epsilon(w_i)} f(z) \, \mathrm{d}z\] for \(\epsilon=1/2\). Hint: Do not attempt the integral directly; compute the relevant Laurent coefficient. It can be useful to use the Taylor expansion \[\frac{1}{a - z} = \frac{1}{a} \frac{1}{1 - z/a} = \frac{1}{a} \sum_{n=0}^\infty (z/a)^n.\]