Let f(z) be single-valued and

analytic in the

annulus R

_{1} < |z - z

_{0}| < R

_{2}. For

points in the annulus, f has the convergent

Laurent series
inf
---
f(z) = > a_n (z - z_0)^n
---
n=-inf

Formally, the coefficients a_{n} are found using the formula

1 / f(z)
a_n = ----- | -------------- dz
2pi*i /c (z - z_0)^n+1
(c a positively oriented closed contour around z_0 inside the annulus)

but in practice Laurent series are often found using algebraic tricks and proved using uniqueness.

Notice that R_{1} and R_{2} can be shrunk and grown, respectively, until the edge of the annulus hits a singularity. This explains, for instance, why the Taylor series for the real function f(x) = 1/(1+x^2) is only valid for |x|<1: in the complex plane, 1/(1+z^2) hits singularities at z = +-i.