Suppose $f(z)$ is analytic on $ {z : 0<|z-\alpha|
I think this is pretty straightforward from the definition of pole. When it is a pole of order m then expanding to the Laurent series we get the principal part has m terms with the lowest power being $-m$. Then multiplying by $(z-\alpha)^m $ and taking limit as $z\rightarrow \alpha.$ we see the non zero guy $a_{-m}$.
I am just confused what is there to be rigorous. Isn't this just a definition? Any suggestion will be appreciated.