I'm trying to construct Laurent series around $\displaystyle z = 0$ and $\displaystyle z= \infty$ for $\displaystyle f(z) = 1/{z-2}$ with $\displaystyle z_{0} = 2$ the only pole of this function.

I have $\displaystyle f(z) = \sum_{n=-\infty}^{\infty}a_{n}(z-c)^n$

with $\displaystyle a_{n} = \oint_{\gamma}f(z)/(z-c)^{n+1}dz$

and $\displaystyle \oint_{\gamma}g(z)dz = 2i\pi res_{z_{0}}g(z)$

and $\displaystyle res_{z_{0}}g(z) = \lim_{z \to \infty}(z - z_{0})g(z)$

so am I correct in assuming that around z=c:

$\displaystyle f(z) = \sum_{n=-\infty}^{\infty}a_{n}(z-c)^n = \sum_{n=-\infty}^{\infty} (z - c)^{n}\lim_{z \to z_{0}}f(z)\frac{z - z_{0}}{(z - c)^{n+1}}$

or did I make a mistake?

And how the hell do I evaluate the function around z=infinity?