I'm having trouble solving this problem:
Find the Laurent series for $\frac{e^z}{z(z-1)}$ when |Z|<1
I dont think I can just expand $e^z$ and $\frac{1}{(z-1)}$ and multiply the terms as the next question states:
"Write the coefficients cn of the previous series explicitly, for both regions, when n = 0, ±1, ±2. Hint: when |z| > 1, the coefficients can be expressed as a familiar infinite series."
How am I meant to do this? The best I've found so far is forming $e^z$ as $e*(e^{z-1})$ and dividing its taylor series by z and z-1 to get:
$-e∑\frac{(z-1)^{n-1}}{zn!}$ which I dont think is the right answer, as there is still a Z on the bottom and I dont know how to get rid of it.
Thank-you!
edit: The sum that ive written is from 0 to infinity, I'm searching how to format this correctly just now. Please still help though :)