Riemann Removability Theorem

Statement

Let ${\displaystyle G\subseteq \mathbb {C} }$ be a domain, ${\displaystyle z_{0}\in G}$, and ${\displaystyle f\colon G\setminus {z_{0}}\to \mathbb {C} }$ be holomorphic. Then ${\displaystyle f}$ can be holomorphically extended to ${\displaystyle z_{0}}$ if and only if there exists a neighborhood ${\displaystyle U\subseteq G}$ of ${\displaystyle z_{0}}$ such that ${\displaystyle f}$ is bounded on ${\displaystyle U\setminus {z_{0}}}$.

Proof

Let ${\displaystyle r>0}$ be chosen such that ${\displaystyle {\bar {B}}_{r}(z_{0})\subseteq U}$, and let ${\displaystyle M}$ be an upper bound for ${\displaystyle f}$ on ${\displaystyle U}$.

We consider the Laurent Series of ${\displaystyle f}$ around ${\displaystyle z_{0}}$. It is

${\displaystyle f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-z_{0})^{n},\qquad a_{n}={\frac {1}{2\pi i}}\int _{|w-z_{0}|=r}{\frac {f(w)}{(w-z_{0})^{n+1}}}\,dw}$

Estimating ${\displaystyle a_{n}}$ gives the so-called Cauchy estimates, namely

${\displaystyle {\begin{array}{rl}|a_{n}|&=\displaystyle \left|{\frac {1}{2\pi i}}\int _{|w-z_{0}|=r}{\frac {f(w)}{(w-z_{0})^{n+1}}}\,dw\right|\\&\leq \displaystyle {\frac {1}{2\pi }}\int _{|w-z_{0}|=r}{\frac {|f(w)|}{|w-z_{0}|^{n+1}}}\,|dw|\\&\leq \displaystyle {\frac {1}{2\pi }}\int _{|w-z_{0}|=r}{\frac {M}{r^{n+1}}}\,|dw|\\&=\displaystyle {\frac {M}{r^{n}}}\\\end{array}}}$

For ${\displaystyle n<0}$, it follows that

${\displaystyle |a_{n}|\leq {\frac {M}{r^{n}}}=Mr^{-n}\to 0,\quad r\to 0}$

Thus, ${\displaystyle a_{n}=0}$ for all ${\displaystyle n<0}$, meaning we have ${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}(z-z_{0})^{n}}$, and ${\displaystyle f(z_{0}):=a_{0}}$ is a holomorphic extension of ${\displaystyle f}$ to ${\displaystyle z_{0}}$.