I'm trying to find the Laurent series around $z = 0$ and specify the largest annulus where the expansion is valid. It's for the functions:
$f(z) = 1/((z-a)(z-b)) $, for $a,b, \in \mathbb{C}$
and
$f(z) = z^3 e^{1/z}$
I've tried to look around the internet for help, but the tasks always differ a bit.
For the first function i tried a partial fraction expansion:
$f(z) = \frac{1}{(b-a)(z-b)} + \frac{1}{(a-b)(z-a)} $
and for function two, i think i need to find simpler expansions so i won't have to do a contour integral?
I would love some tips and inputs.
For the first one, you did the right thing. The next step is to write:$$f(z)=\frac1{ab-b^2}\times\frac1{1-\frac zb}+\frac1{ab-a^2}\times\frac1{1-\frac za}.$$On the other hand$$(\forall z\in\mathbb C\setminus\{0\}):z^3e^{\frac1z}=z^3+z^2+\frac z{2!}+\frac1{3!}+\frac1{4!z}+\cdots$$