# Complex Variables: Poles

1. Sep 9, 2009

### Winzer

1. The problem statement, all variables and given/known data
Determine the nature of the singularity at z=0

2. Relevant equations
$f(z)=\frac{1}{cos(z)}+\frac{1}{z}$

3. The attempt at a solution
by expanding into series:

$f(z)=\Sigma_{n=0}^{\infty} \frac{(2n)! (-1)^n}{x^{2n}} + \Sigma_{n=0}^{\infty} (-1)^n (z-1)^n$
Now $\frac{1}{z}$ has no principle part, [tex] b_m=0[/itex].
This leaves the only principle part from cos. $b_m=(2m)! (-1)^m$. There are infinite bm
so the behaviour is an essential singularity.