# Complex Analysis/Local theory of holomorphic functions

Proposition (holomorphic function has Taylor expansion convergent on its domain):

Let ${\displaystyle U\subseteq \mathbb {C} }$ be open, and let ${\displaystyle f:U\to \mathbb {C} }$ be holomorphic. Let ${\displaystyle z_{0}\in U}$. Then there exists coefficients ${\displaystyle a_{0},a_{1},\ldots ,a_{n},\ldots }$ of a Taylor series such that whenever ${\displaystyle {\overline {B_{r}(z_{0})}}\subseteq \mathbb {C} }$ is a ball of radius ${\displaystyle r>0}$ that is contained within ${\displaystyle U}$, then the Taylor series

${\displaystyle \sum _{n=0}^{\infty }a_{n}(z-z_{0})^{n}}$

converges absolutely on ${\displaystyle {\overline {B_{r}(z_{0})}}}$ and equals ${\displaystyle f(z)}$ there.

Proof: By Cauchy's formula, we have

${\displaystyle f(z)=\int _{\partial B_{r}(z_{0})}{\frac {f(w)}{w-z}}dw}$.

Adding a zero, we may rewrite this as

${\displaystyle f(z)=\int _{\partial B_{r}(z_{0})}{\frac {f(w)}{w-z_{0}-z-z_{0}}}dw}$.

Computing further, and using the convergence of the geometric series for arguments of modulus ${\displaystyle <1}$,

{\displaystyle {\begin{aligned}f(z)&=\int _{\partial B_{r}(z_{0})}{\frac {f(w)}{w-z_{0}-z-z_{0}}}dw\\&=\int _{\partial B_{r}(z_{0})}{\frac {f(w)}{1-{\frac {z-z_{0}}{w-z_{0}}}}}{\frac {1}{w-z_{0}}}dw\\&=\int _{\partial B_{r}(z_{0})}{\frac {f(w)}{w-z_{0}}}\sum _{n=0}^{\infty }\left({\frac {z-z_{0}}{w-z_{0}}}\right)^{n}dw\\&=\sum _{n=0}^{\infty }\underbrace {\int _{\partial B_{r}(z_{0})}{\frac {f(w)}{(w-z_{0})^{n+1}}}dw} _{=:a_{n}}(z-z_{0})^{n};\end{aligned}}}

here, interchanging differentiation and integration is justified by the absolute convergence of the former expression. ${\displaystyle \Box }$

Theorem (identiy theorem):

Let ${\displaystyle U\subseteq \mathbb {C} }$ be open, and let ${\displaystyle f,g:U\to \mathbb {C} }$ be two holomorphic functions defined on ${\displaystyle U}$. Suppose that the point ${\displaystyle z_{0}\in U}$ is an accumulation point of the set ${\displaystyle \{z\in U|f(z)=g(z)\}}$. Then ${\displaystyle f(z)=g(z)}$ for all ${\displaystyle z}$ in the connected component of ${\displaystyle z_{0}}$ of ${\displaystyle U}$.

Proof: We develop both ${\displaystyle \Box }$

Theorem (Riemann's theorem on removable singularities):

Let ${\displaystyle f:U\to \mathbb {C} }$ be a function which is holomorphic on ${\displaystyle U\setminus \{z_{0}\}}$ for a certain ${\displaystyle z_{0}\in U}$, and furthermore bounded in a ball (or rather disk) about ${\displaystyle z_{0}}$, say ${\displaystyle B_{r}(z_{0})\subseteq U}$, where ${\displaystyle r>0}$ is just the radius of that small ball. Then we will find a value ${\displaystyle w_{0}}$ such that if we continue ${\displaystyle f}$ to ${\displaystyle w_{0}}$ at the point ${\displaystyle z_{0}}$, the result will be holomorphic.

Proof: We define the new function

${\displaystyle g:U\to \mathbb {C} ,h(z)={\begin{cases}(z-z_{0})^{2}f(z)&z\neq z_{0}\\0&z=z_{0}\end{cases}}}$

and claim that it's holomorphic on ${\displaystyle U}$. It will be complex differentiable in ${\displaystyle U\setminus \{z_{0}\}}$ "by the product rule" (which is supposed here to include the statement that the product of complex differentiable functions is complex differentiable), and in ${\displaystyle z_{0}}$ we have

${\displaystyle \lim _{g\to 0}{\frac {g(z_{0}+h)-g(z_{0})}{h}}={\frac {h^{2}(f(z_{0}+h)-f(z))}{h}}=0}$

since ${\displaystyle h}$ is bounded.

Now we develop ${\displaystyle h}$ into a Taylor series at ${\displaystyle z_{0}}$. But the first two coefficients of it will be zero, whence ${\displaystyle h}$ will be divisible by ${\displaystyle (z-z_{0})^{2}}$ to obtain a power series, and this power series will define a holomorphic function about ${\displaystyle z_{0}}$. But by the definition of ${\displaystyle h}$, this series coincides with ${\displaystyle f}$ everywhere except ${\displaystyle z_{0}}$, whence if ${\displaystyle f}$ is continued at ${\displaystyle z_{0}}$ by the constant term of the series, the result will be holomorphic and thus the desired continuation. ${\displaystyle \Box }$

smoothness, integrability