Complex Analysis- Singularities

[TheForumLord]

Homework Statement

Let f be analytic at the complex plane excapt for z= -1 and z=3 which are simple poles of f.

Let \Sigma_{-\infty}^{-1} a_{n}(z-2)^{n} be the Laurent series of f.
In part A I've found that the series converges at 1<|z-2|<3 .
B is: Find the coeefficients a_{n} of the given Laurent series.
Hint: Look at g(z) = (z+1)(z-3)f(z)

Homework Equations

The Attempt at a Solution

We know that g(z) has no poles or singularities whatsoever. So Laurent series of g is actually a Taylor series... But how can we find from this data the given coeefficients?

Thanks in advance

 

[vela]

Are you sure that's a correct Laurent series for f(z)? Doesn't it have an essential singularity at z=2?

 

[TheForumLord]

I'm sure indeed...I had no typos in this one... But the an's can be also zero or something...