# Complex Analysis: Finding Laurent Series in given region.

I am struggling with 23B, in particular for getting the series to converge for $|z|<1$. I did partial fractions decomposition and was able to break up the series into powers of $(z+3)(z-3)(z+1)$. For the fraction of $z+1$, I can rewrite that as a geometric series which converges for $|z|<1$. However, for the other two powers,$(z+3)$ and $(z-3)$, if I rewrite them as geometric series, I get that they converge for $|z|<3$. However, I need the whole series to be valid/converge for $|z|<1$. I am confused/not sure how to do this. Any tips/pointers/suggestions/help? I'm just stuck on what to do. The image of the question is attached.

Laurent Series

## 1 Answer

Let's tackle $\dfrac{1}{z+3}$. For the region $|z|<1$, you can simply write it as $\dfrac{1}{3}\cdot\dfrac{1}{1-(-\frac{z}{3})}$ and now you can use the power seris expansion because $$\Big|\dfrac{z}{3}\Big|<\dfrac{1}{3}<1$$.

You can use the exact same idea to do the remaining fractions for the other region too. Only the algebra will be slightly different.