- #1

skrat

- 748

- 8

## Homework Statement

Calculate ##\int _Kz^2exp(\frac{2}{z})dz## where ##K## is unit circle.

## Homework Equations

## The Attempt at a Solution

Hmmm, I am having some troubles here. Here is how I tried:

In general ##\int _\gamma f(z)dz=2\pi i\sum_{k=1}^{n}I(\gamma,a_k)Res(f,a_k)## where in my case ##I(\gamma , a_k)=1##.

Now ##Res(f,a_k)##:

##f(z)=z^2exp(\frac{2}{z})=z^2\sum_{n=0}^{\infty}\frac{1}{n!}(\frac{2}{z})^n=\sum_{n=0}^{\infty}\frac{1}{n!}\frac{2^n}{z^{n-2}}##

To get ##\frac{1}{z}## clearly ##n-2=1## so ##n=3##.

Which gives me ##Res(f , 0)=\frac{4}{3}##.

Therefore ##\int _Kz^2exp(\frac{2}{z})dz=2\pi i \frac{4}{3}##.Or is that completely wrong? Thanks in advance!