= Solution
<Laurent's theorem> states that a function holomorphic on $R<|z-a|<S$ has a unique expansion
$$
f(z)=\sum_{n=-\infty}^{\infty}c_n(z-a)^n,
$$
converging uniformly on every compact subannulus. Its coefficients are
$$
\boxed{c_n=\frac1{2\pi i}\int_{|\zeta-a|=\rho}\frac{f(\zeta)}{(\zeta-a)^{n+1}}\,d\zeta,\qquad R<\rho<S,}
$$
with counterclockwise orientation. The value is independent of the chosen intermediate radius. The nonnegative powers form a power series convergent inside the outer radius, and the negative powers form the <principal part of a Laurent series>.
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