= Solution
On an annulus $r<|z-z_0|<R$, the <Laurent theorem> represents an analytic function uniquely as
$$
f(z)=\sum_{n=-\infty}^{\infty}a_n(z-z_0)^n.
$$
For any positively oriented simple closed contour $C$ in the annulus winding once around $z_0$, the coefficients in the <Laurent series> are
$$
\boxed{a_n=\frac1{2\pi i}\oint_C
\frac{f(\zeta)}{(\zeta-z_0)^{n+1}}\,d\zeta.}
$$
The formula follows from the <Cauchy integral formula> applied term by term, and it is independent of the chosen contour by the <Cauchy integral theorem>.
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