Laurent's theorem
Theorem 9.1 (Laurent’s theorem). If \(f\) is analytic in the annulus \(r < \abs {z - z_0} < R\), then
\[ f(z) = \sum _{n=-\infty }^{\infty } a_n (z - z_0)^n \]
there, where both halves of the series converge absolutely in the annulus and uniformly on every closed subannulus. The coefficients are unique, and
\[ a_n = \frac {1}{2\pi i} \oint _C \frac {f(\zeta )}{(\zeta - z_0)^{n+1}} \dd \zeta \]
for any positively oriented circle \(C \colon \abs {\zeta - z_0} = \rho \) with \(r < \rho < R\).
Source: [1, ch. 5]