15 - Laurent Series

Laurent Series in a Punctured Disc corollary

If $f:U\to \mathbb{C}$ is a holomorphic function and $z_0$ is an isolated singularity then

$$f\text{ has a Laurent Expansion on punctured disc }B(z_0,R)\setminus \{x_0\}$$

for any $R$ such that $B(z_0,R)\setminus \{z_0\}\subset U$

Proof

Let $r$ be such that $0<r<R$ and apply Laurent’s Theorem to $A=A(z_0,r,R)$
As the coefficients $c_n$ can be written in terms of integrals along ${\gamma }_R$ then

$${\gamma }_R\text{ is independent of }r$$

By sending $r$ to zero then Laurent Series converges in

$$B(z_0,R)=\bigcup _{o<r<R}A(z_0,r,R)$$

and converges uniformly in $A(z_0,r,R^{\prime })$ for any $0<r<R^{\prime }<R$

Principal Part of the Laurent's Series

Let $z_0$ be an isolated singularity of $f$ with Laurent’s Expansion

$$\sum _{n=-\infty }^{\infty }{c}_n(z-z_0{)}^n$$

Principal Part of $f$ at $z_0$ is the sum of terms with negative powers denoted by

$$P_{z_0}f(z)=\sum _{n=-\infty }^{-1}{c}_n(z-z_0{)}^n=\sum _{n=1}^{\infty }{c}_{-n}(z-z_0{)}^{-n}$$

Convergence of the Principal Part

Principal Part of $f$ at $z_0$ converges on $\mathbb{C}\setminus \{z_0\}$ and converges uniformly on $\mathbb{C}\setminus B(z_0,r)$

Proof

Follows from proof of Laurent’s Theorem
f $f$ is holomorphic on $A(z_0,r,R)$ then principal part converges uniformly on

$$\{z:|z-z_0|>r^{\prime }\}\text{ for any }{r}^{\prime }>r$$

Since for isolated singularities, $r$ can be arbitrarily small then claim follows immediately