Laurent series

Consider $0 \leq r_{1} < r_{2} \leq \infty$ , and let $z_{1} \in C$ . Suppose $f \in H (Ω)$ , where

Ω = {z \in C : r_{1} < | z - z_{1} | < r_{2}} .

Then

f (z) = \sum_{n = 0}^{\infty} a_{n} (z - z_{1})^{n} + \sum_{n = 1}^{\infty} \frac{b_{n}}{(z - z_{1})^{n}}

where both series converge absolutely in $Ω$ , and uniformly on compact subsets of the form

{z \in C : ρ_{1} \leq | z - z_{1} | \leq ρ_{2}} with r_{1} < ρ_{1} < ρ_{2} < r_{2} .

Moreover, the coefficients are given by:

a_{n} = \frac{1}{2 π i} \int_{γ} \frac{f (z)}{(z - z_{1})^{n + 1}} d z, b_{n} = \frac{1}{2 π i} \int_{γ} f (z) (z - z_{1})^{n - 1} d z,

where the contour $γ$ is parametrized by

γ (t) = z_{1} + r e^{i t}, t \in [0, 2 π], with r_{1} < r < r_{2} .