Laurent series

A Laurent series centered about $a$ is a series of the form

$$\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_k{(z-a)}^k$$

where $c_k,a,z\in ℂ$. The principal part of a Laurent series is the subseries ${\sum }_{k=-\mathrm{\infty }}^{-1}{c}_k{(z-a)}^k$.

One can prove that the above series converges everywhere inside the (possibly empty) set

where

$$R_1:=\underset{k\to \mathrm{\infty }}{lim\; sup}{|c_{-k}|}^{1/k}$$

and

$$R_2:=1/\left(\underset{k\to \mathrm{\infty }}{lim\; sup}{|c_k|}^{1/k}\right).$$

Every Laurent series has an associated function, given by

$$f(z):=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_k{(z-a)}^k,$$

whose domain is the set of points in $ℂ$ on which the series converges. This function is analytic inside the annulus $D$, and conversely, every analytic function on an annulus is equal to some unique Laurent series. The coefficients of Laurent series for an analytic function can be determined using the Cauchy integral formula.