Complex Analysis/Global theory of holomorphic functions

Theorem (Liouville's theorem):

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

$\forall z\in \mathbb {C} \setminus B_{R}(0):\|f(z)\|<M|z^{n}|$,

then $f$ is a polynomial of degree less than or equal to $n$.

Proof 1: First note that

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

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

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

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

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

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

Theorem (identity theorem):

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

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

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

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

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

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

Theorem (maximum principle):

Theorem (argument principle):

Theorem (Rouché's theorem):

Theorem (Hurwitz's theorem):

Theorem (Hartog's extension theorem):

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

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

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

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

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

Theorem (Weierstraß preparation theorem):

Exercises

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