2 Classifying singularities

If the function $f\left(z\right)$ has a singularity at $z=z_0$ , and in a neighbourhood of $z_0$ (i.e. a region of the complex plane which contains $z_0$ ) there are no other singularities, then $z_0$ is an isolated singularity of $f\left(z\right)$ .

The principal part of the Laurent series is the part containing negative powers of $\left(z-z_0\right)$ . If the principal part has a finite number of terms say

$\frac{b_1}{z-z_0}+\frac{b_2}{{\left(z-z_0\right)}^2}+\dots +\frac{b_m}{{\left(z-z_0\right)}^m}$ and $b_m\ne 0$

then $f\left(z\right)$ has a pole of order $m$ at $z=z_0$ (we have written $b_1$ for $a_{-1},$ $b_2$ for $a_{-2}$ etc. for simplicity.) Note that if $b_1=b_2=\dots =0$ and $b_m\ne 0$ , the pole is still of order $m$ .

A pole of order 1 is called a simple pole whilst a pole of order 2 is called a double pole . If the principal part of the Laurent series has an infinite number of terms then $z=z_0$ is called an isolated essential singularity of $f\left(z\right)$ .

The function

$f\left(z\right)=\frac{i}{z\left(z-i\right)}\equiv \frac{1}{z-i}-\frac{1}{z}$

has a simple pole at $z=0$ and another simple pole at $z=i$ . The function $e^{\frac{1}{z-2}}$ has an isolated essential singularity at $z=2$ . Some complex functions have non-isolated singularities called branch points . An example of such a function is $\sqrt{z}$ .

Task!

Classify the singularities of the function $f\left(z\right)=\frac{2}{z}-\frac{1}{z^2}+\frac{1}{z+i}+\frac{3}{{\left(z-i\right)}^4}$ .

Answer

Exercises

1. Expand  $f\left(z\right)=\frac{1}{2-z}$  in terms of negative powers of $z$ to give a series which will be valid if $\left|z\right|>2$ .
2. Classify the singularities of the function: $f\left(z\right)=\frac{1}{z^2}+\frac{1}{{\left(z+i\right)}^2}-\frac{2}{{\left(z+i\right)}^3}$ .

Answer