How to find the Laurent series for a function

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I'm trying to understand how to find a Laurent series for a function.

The Laurent series is given by $$f(z) = \sum^\infty_{n =-\infty} \frac{1}{2\pi i} \oint_C \frac{f(\epsilon)}{(\epsilon -a)^{n+1}}d\epsilon (z-a)^n$$

However, every example I found seems not to use this formula.

For $$f(z) = \frac{sinz}{z^2}$$ at the origin

How it works? Should I just plug a = 0 then integrate f(z). I'm really not sure how it works.