Let f(z) be single-valued and analytic
in the annulus
< |z - z0
| < R2
. For points
in the annulus, f has the convergent Laurent series
f(z) = > a_n (z - z_0)^n
Formally, the coefficients an 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 R1 and R2 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.