7.7 Isolated singularities, and the residue theorem  (Page 3/4)

Let $f$ be continuous on a punctured disk ${\overline{B^{\text{'}}}}_r\left(c\right),$ analytic at each point of $B_r^{\text{'}}\left(c\right),$ and suppose that $c$ is an essential singularity of $f.$ Then

1. For all $z\in B_r^{\text{'}}\left(c\right),$
   $$f\left(z\right)=\sum _{k=-\infty }^{\infty }{a}_k{(z-c)}^k,$$
   where the sequence ${\left\{a_k\right\}}_{-\infty }^{\infty }$ has the property that for any negative integer $N$ there is a $k<N$ such that $a_k\ne 0.$
2. The infinite series in part (1) converges uniformly on each compact subset $K$ of $B_r^{\text{'}}\left(c\right).$ That is, if $F_n$ is defined by $F_n\left(z\right)={\sum }_{k=-n}^n{a}_k{(z-c)}^k,$ then the sequence $\left\{F_n\right\}$ converges uniformly to $f$ on the compact set $K.$
3. For any piecewise smooth geometric set $S\subseteq B_r\left(c\right),$ whose boundary $C_S$ has finite length, and satisfying $c\in S^0,$ we have
   $${\int }_{C_S}f\left(\zeta \right)d\zeta =2\pi i{a}_{-1},$$
   where $a_{-1}$ is the coefficient of ${(z-c)}^{-1}$ in the series of part (1).

Define numbers ${\left\{a_k\right\}}_{-\infty }^{\infty }$ as follows.

Note that for any $0<\delta <r$ we have from Cauchy's Theorem that

where $C_{\delta }$ denotes the boundary of the disk ${\overline{B}}_{\delta }\left(c\right).$

Let $z\ne c$ be in $B_r\left(c\right),$ and choose $\delta >0$ such that $\delta <|z-c|.$ Then, using part (c) of [link] , and then mimicking the proof of [link] , we have

which proves part (1).

We leave the proofs of parts (2) and (3) to the exercises.

REMARK The representation of $f\left(z\right)$ in the punctured disk $B_r^{\text{'}}\left(c\right)$ given in part (1) of [link] and [link] is called the Laurent expansion of $f$ around the singularity $c.$ Of course it differs from a Taylor series representation of $f,$ as this one contains negative powers of $z-c.$ In fact, which negative powers it contains indicates what kind of singularity the point $c$ is.

Non removable isolated singularities of a function $f$ share the property that the integral of $f$ around a disk centered at the singularity equals $2\pi i{a}_{-1},$ where the number $a_{-1}$ is the coefficient of ${(z-c)}^{-1}$ in the Laurent expansion of $f$ around $c.$ This number $2\pi i{a}_{-1}$ is obviously significant, and we call it the residue of f at c, and denote it by $R_f\left(c\right).$

Combining [link] , [link] , and [link] , we obtain:

Residue theorem

Let $S$ be a piecewise smooth geometric set whose boundary has finite length, let $c_1,...,c_n$ be points in $S^0,$ and suppose $f$ is a complex-valued function that is continuous at every point $z$ in $S$ except the $c_k$ 's, and differentiable at every point $z\in S^0$ except at the $c_k$ 's. Assume finally that each $c_k$ is a nonremovable isolated singularity of $f.$ Then

That is, the contour integral around $C_S$ is just the sum of the residues inside $S.$