Complex Analysis/Global theory of holomorphic functions

Template:Theorem

Exercises

1. Use Liouville's theorem to demonstrate that every non-constant polynomial $p\in ℂ[z_1,\dots ,z_n]$ has at least one root in ${ℂ}^n$ (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 $f(x_1,\dots ,x_k)=\sum _{\alpha \in {\mathbb{N}}^k}^{\mathrm{\infty }}{a}_{\alpha }(x_1,\dots ,x_k{)}^{\alpha }$ be a power series with real coefficients which converges absolutely on an open neighbourhood of the origin of ${ℝ}^n$. Prove that $f$ may be extended to a function on an open neighbourhood of the origin of the complex plane.
   2. Let $g(x)=\sum _{n=0}^{\mathrm{\infty }}{b}_n{x}^n$ 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 $x\in ℝ$ st. $|x|<R$, where $R>0$ is a real number. Prove that $g$ may be extended to a holomorphic function on $B_R(0)\subset ℂ$.
   3. Prove that the extensions considered in the first two sub-exercises are unique.
3. Let $f:ℂ\to ℂ$ be an entire function and let $0\le \alpha <1$, $k\in \mathbb{N}$ and $C>0$ such that $\mathrm{\forall }z\in ℂ:|f(z)|\le C(1+|z{|}^{k+\alpha })$. Prove that $f$ is a polynomial of degree $\le k$.