7.7 Isolated singularities, and the residue theorem  (Page 2/4)

Since this holds for arbitrary $ϵ>0,$ we see that $\tilde{f}\left(z\right)=F\left(z\right)$ for all $z\ne c$ in $B_r\left(c\right).$

Finally, since

the equality of $F$ and $\tilde{f}$ on all of $B_r\left(c\right)$ is proved. This finishes the proof of part (1).

Now, for the second kind of isolated singularity:

A complex number $c$ is called a pole of a function $f$ if there exists an $r>0$ such that $f$ is continuous on the punctured disk ${\overline{B^{\text{'}}}}_r\left(c\right),$ analytic at each point of $B_r^{\text{'}}\left(c\right),$ the point $c$ is not a removable singularity of $f,$ and there exists a positive integer $k$ such that the analytic function ${(z-c)}^{k}f\left(z\right)$ has a removable singularity at $c.$

A pole $c$ of $f$ is said to be of order n, if $n$ is the smallest positive integer for which the function $\tilde{f}\left(z\right)\equiv {(z-c)}^{n}f\left(z\right)$ has a removable singularity at $c.$

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 a pole of order $n$ of $f.$ Then

1. For all $z\in B_r^{\text{'}}\left(c\right),$
   $$f\left(z\right)=\sum _{k=-n}^{\infty }{a}_k{(z-c)}^k.$$
2. The infinite series of part (1) converges uniformly on each compact subset $K$ of $B_r^{\text{'}}\left(c\right).$
3. For any piecewise smooth geometric set $S\subseteq B_r\left(c\right),$ whose boundary $C_S$ has finite length, and satisfying $c\in S^0,$
   $${\int }_{C_S}f\left(\zeta \right)d\zeta =2\pi i{a}_{-1},$$
   where $A_{-1}$ is the coefficient of ${(z-c)}^{-1}$ in the series of part (1).

For each $z\in B_r^{\text{'}}\left(c\right),$ write $\tilde{f}\left(z\right)={(z-c)}^{n}f\left(z\right).$ Then, by [link] , $\tilde{f}$ is analytic on $B_r\left(c\right),$ whence

where $a_k=c_{n+k}.$ This proves part (1).

We leave the proof of the uniform convergence of the series on each compact subset of $B_r^{\text{'}}\left(c\right),$ i.e., the proof of part (2), to the exercises.

Part (3) follows from Cauchy's Theorem ( [link] ) and the computations in [link] . Thus:

as desired. The summation sign comes out of the integral because of the uniform convergence of the series on the compact circle $C_r.$

Having defined two kinds of isolated singularities of a function $f,$ the removable ones and the polls of finite order, there remain all the others, which we collect into a third type.

Let $f$ be continuous on a punctured disk ${\overline{B^{\text{'}}}}_r\left(c\right),$ and analytic at each point of $B_r^{\text{'}}\left(c\right).$ The point $c$ is called an essential singularity of $f$ if it is neither a removable singularity nor a poll of any finite order. Singularities that are either poles or essential singularities arecalled nonremovable singularities.