17 - Characterisation of Isolated Singularities

Characterisation of Isolated Singularities

Let $z_0$ be an isolated singularity of $f$
Let ${\sum }_{n=-\infty }^{\infty }{c}_n(z-z_0{)}^n$ be the Laurent Expansion of $f$ then

$z_0$ is classified as

1. Removable Singularity: If $c_n=0$ for all $n<0$

2. Pole of Order $n$: If $c_{-n}\ne 0$ and $c_k=0$ for all $k<n$

3. Essential Singularity: There is arbitrary large $n$ such that $c_n\ne 0$

Equivalent Statements for Principal

1. Principal Part vanishes

2. Principal Part is non-trivial but contains a finite number of non-zero terms

3. Principal Part contains infinitely many non-zero terms

Proof

If there is no principal part then $f$ has a ordinary power series for it’s Laurent Expansion
$f$ would converge to function $g$ which is analytic in some disc $B(z_0,R)$ around $z_0$
where $f(z)=g(z)$ for all $0<|z|<R$ so $z_0$ is removable

If $z_0$ is removable then there is function $g$ which is holomorphic at $z_0$
with $g$ coinciding with $f$ in $B(z_0,R)\setminus \{z_0\}$
Hence Laurent Expansion coefficients are equal, but for holomorphic function $g$ then
All coefficients with $n<0$ are given by integrals of holomorphic function $(z-z_0{)}^{-n}g(z)$
However by Cauchy’s Theorem for Simply Connected Domains then integrals vanish

If Laurent Expansion is in form

$$f(z)=\frac{c_{-n}}{(z-z_0{)}^n}+\cdots +c_0+c_1(z-z_0)+\cdots$$

with $c_{-n}\ne 0$ then $f(z)=(z-z_0{)}^{-n}g(z)$ where

$$g(z)=c_{-n}+c_{-n+1}(z-z_0)+\cdots$$

which is analytic in some neighbourhood of $z_0$

If $z_0$ is a pole of order $n$ then there is holomorphic $g$ such that

$$g(z_0)\ne 0\text{ and }f(z)=(z-z_0{)}^{-n}g(z)$$

Writing the Taylor Series of $g$ then the Laurent Expansion of $f$ is in form

$$f(z)=\frac{g(z_0)}{(z-z_0{)}^n}+\frac{g^{\prime }(z_0)}{(z-z_0{)}^{n-1}}\cdots$$

where the last part follows trivially from the last two