Laurent series (idea)

Let f(z) be single-valued and analytic in the annulus R₁ < |z - z₀| < R₂. 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 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.