# Complex Analysis/Global theory of holomorphic functions

Theorem (Liouville's theorem):

Let ${\displaystyle X}$ be a Banach space, and let ${\displaystyle f:\mathbb {C} ^{n}\to X}$ be an entire function. If there exists a natural number ${\displaystyle n\in \mathbb {N} }$ and two constants ${\displaystyle M,R>0}$ such that

${\displaystyle \forall z\in \mathbb {C} \setminus B_{R}(0):\|f(z)\|,

then ${\displaystyle f}$ is a polynomial of degree less than or equal to ${\displaystyle n}$.

Proof 1: First note that

${\displaystyle \lim _{w\to \infty }{\frac {|z-w|}{|w|}}=1}$.

Let ${\displaystyle z\in \mathbb {C} }$ and ${\displaystyle r>\max\{R,|z|\}}$. Then Cauchy's integral formula and the triangle inequality for integrals together imply that

${\displaystyle |f^{(n+1)}(z)|\leq {\frac {1}{2\pi n!}}\int _{\partial B_{r}(0)}{\frac {\|f(w)\|}{|w-z|^{n+2}}}dw\leq {\frac {C}{2\pi n!}}\int _{\partial B_{r}(0)}{\frac {1}{r^{2}}}dz}$

for a certain ${\displaystyle C>0}$. The latter expression may be computed explicitly; it equals

${\displaystyle {\frac {C}{n!r}}}$,

which tends to zero as ${\displaystyle r\to \infty }$. Hence, ${\displaystyle f^{(n+1)}}$ vanishes and ${\displaystyle f}$ is a polynomial of degree ${\displaystyle \leq n}$. ${\displaystyle \Box }$

Theorem (identity theorem):

Let ${\displaystyle X}$ be a Banach space, let ${\displaystyle U\subseteq \mathbb {C} }$ be open and connected, let ${\displaystyle z_{0}\in U}$ and let ${\displaystyle f,g:U\to X}$ be two holomorphic functions on ${\displaystyle U}$ such that the set ${\displaystyle \{z\in U|f(z)=g(z)\}}$ has a cluster point ${\displaystyle z_{0}\in U}$. Then ${\displaystyle f=g}$.

Proof: Let ${\displaystyle w_{0}\in U}$ be any point. Since holomorphic functions are analytic, the function ${\displaystyle f-g}$ posesses a power series expansion

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

which converges on a sufficiently small neighbourhood of ${\displaystyle w_{0}}$.

Suppose first that ${\displaystyle w_{0}}$ is a cluster point of the set ${\displaystyle \{z\in U|f(z)=g(z)\}}$.

Let ${\displaystyle n_{0}\in N}$ be the least natural number such that ${\displaystyle a_{n_{0}}\neq 0}$. ${\displaystyle \Box }$

Theorem (maximum principle):

Theorem (argument principle):

Theorem (Rouché's theorem):

Theorem (Hurwitz's theorem):

Theorem (Hartog's extension theorem):

Let ${\displaystyle {\vec {\epsilon }}>{\vec {\delta }}}$, and let ${\displaystyle f:B_{\vec {\epsilon }}(0)\setminus B_{\vec {\delta }}(0)\to \mathbb {C} }$ be holomorphic, where ${\displaystyle B_{\vec {\epsilon }}(0)\subseteq \mathbb {C} ^{n}}$ with ${\displaystyle n\geq 2}$. Then there exists a unique function ${\displaystyle F:B_{\vec {\epsilon }}(0)\to \mathbb {C} }$ such that

${\displaystyle F|_{B_{\vec {\epsilon }}(0)\setminus B_{\vec {\delta }}(0)}=f}$.

Proof: Since ${\displaystyle n\geq 2}$, we may pick the following subset of ${\displaystyle B_{\vec {\epsilon }}(0)\setminus B_{\vec {\delta }}(0)}$:

${\displaystyle W=\{(z_{1},\ldots ,z_{n})|\epsilon _{1}-\alpha <|z_{1}|<\epsilon _{1}\wedge \forall j\in \{2,\ldots ,n\}:|z_{j}|<\epsilon _{1}\}\cup \{\}}$,

where ${\displaystyle \alpha >0}$ is sufficiently small. Since the restriction of a holomorphic function is holomorphic, ${\displaystyle f}$ is holomorphic on ${\displaystyle W}$. Moreover, ${\displaystyle \Box }$

Theorem (Weierstraß preparation theorem):

## Exercises

1. Use Liouville's theorem to demonstrate that every non-constant polynomial ${\displaystyle p\in \mathbb {C} [z_{1},\ldots ,z_{n}]}$ has at least one root in ${\displaystyle \mathbb {C} ^{n}}$ (Hint: Consider the function ${\displaystyle 1/p}$).
2. In this exercise, we want to look at the simplest sufficient conditions for the possibility of extending a function given by a real power series to a function on the complex plane.
1. Let ${\displaystyle f(x_{1},\ldots ,x_{k})=\sum _{\alpha \in \mathbb {N} ^{k}}^{\infty }a_{\alpha }(x_{1},\ldots ,x_{k})^{\alpha }}$ be a power series with real coefficients which converges absolutely on an open neighbourhood of the origin of ${\displaystyle \mathbb {R} ^{n}}$. Prove that ${\displaystyle f}$ may be extended to a function on an open neighbourhood of the origin of the complex plane.
2. Let ${\displaystyle g(x)=\sum _{n=0}^{\infty }b_{n}x^{n}}$ be a power series such that for all ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle b_{n}}$ is real and positive. Suppose further that ${\displaystyle g}$ converges for all ${\displaystyle x\in \mathbb {R} }$ st. ${\displaystyle |x|, where ${\displaystyle R>0}$ is a real number. Prove that ${\displaystyle g}$ may be extended to a holomorphic function on ${\displaystyle B_{R}(0)\subset \mathbb {C} }$.
3. Prove that the extensions considered in the first two sub-exercises are unique.
3. Let ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$ be an entire function and let ${\displaystyle 0\leq \alpha <1}$, ${\displaystyle k\in \mathbb {N} }$ and ${\displaystyle C>0}$ such that ${\displaystyle \forall z\in \mathbb {C} :|f(z)|\leq C(1+|z|^{k+\alpha })}$. Prove that ${\displaystyle f}$ is a polynomial of degree ${\displaystyle \leq k}$.