Additional Problems for Chapter 9 \( \newcommand{\ds}{\displaystyle} \newcommand{\diam}{\operatorname{diam}} \newcommand{\area}{\operatorname{area}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\res}{\operatorname{res}} \newcommand{\wind}{\operatorname{W}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\iso}{\stackrel{\cong}{\longrightarrow}} \newcommand{\myint}{\operatorname{int}} \newcommand{\ve}{\varepsilon} \newcommand{\es}{\emptyset} \newcommand{\sm}{\smallsetminus} \newcommand{\bd}{\partial} \newcommand{\Chat}{\hat{\mathbb C}} \newcommand{\Cstar}{{\mathbb C}^*} \newcommand{\Dstar}{{\mathbb D}^*} \newcommand{\myre}{\operatorname{Re}} \newcommand{\myim}{\operatorname{Im}} \newcommand{\ov}{\overline} \newcommand{\io}{\iota} \newcommand{\con}{\operatorname{const.}} \newcommand{\OO}{{\mathcal O}} \newcommand{\MM}{{\mathcal M}} \newcommand{\FF}{{\mathcal F}} \newcommand{\CC}{{\mathbb C}} \newcommand{\PP}{{\mathbb P}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HH}{{\mathbb H}} \newcommand{\TT}{{\mathbb T}} \newcommand{\II}{{\mathbb I}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\DD}{{\mathbb D}} \newcommand{\QQ}{{\mathbb Q}} \)




Additional Problems for Chapter 9

  1. If $f,g \in \OO(\CC)$ have no common zeros, show that there exist $F,G \in \OO(\CC)$ such that $fF+gG=1$ everywhere in $\CC$. Thus, the ideal generated by $f,g$ is the whole ring $\OO(\CC)$. (Hint: At every zero of $g$ of order $n$, the function $1-fF$ must have a zero of order at least $n$.)
  2. Construct an $f \in \OO(\DD)$ with the property that $f(\DD \cap \DD(p,\ve))$ is dense in $\CC$ for every $p \in \TT$ and every $\ve>0$. (Hint: Construct sequences $\{ z_n \}$ in $\DD$ and $\{ w_n \}$ in $\CC$ such that for every $p \in \TT$ and every $q \in \CC$ there is an increasing sequence $n_k$ of integers with $z_{n_k} \to p$ and $w_{n_k} \to q$. Then use the interpolation Theorem 9.5.)
  3. Show that there is a sequence $\{ P_n \}$ of complex polynomial maps such that \[ \lim_{n \to \infty} P_n(z) = \begin{cases} e^z & z \in \RR \\ 0 & z \in \CC \sm \RR. \end{cases} \]
  4. Suppose $D:\OO(U) \to \OO(U)$ is a derivation, i.e., $D$ is a continuous linear operator on $\OO(U)$ which satisfies the product rule $D(fg)=D(f)g+fD(g)$ for all $f,g \in \OO(U)$. Show that $D(f)=f' D(z)$ for all $f \in \OO(U)$. In particular, if $D(z)=1$, then $D$ is the differentiation operator. (Hint: Prove the result for polynomials and then rational functions with poles outside $U$. Then apply Runge's theorem.)
  5. Let $K_1$ and $K_2$ be disjoint compact connected sets in $\CC$ such that $U:=\CC \sm (K_1 \cup K_2)$ is connected. Show that every $f \in \OO(U)$ can be decomposed as $f=f_1+f_2$, where $f_j \in \OO(\CC \sm K_j)$. Generalize to finitely connected domains. (Hint: Use a standard basis $\langle \gamma_1 \rangle, \langle \gamma_2 \rangle$ for $H_1(U)$ as in $\S 9.4$, where the image $|\gamma_j|$ can be taken arbirtrarily close to $K_j$. Apply Cauchy's integral formula to the cycle $\TT(0,r)-\gamma_1-\gamma_2$ for large $r>0$.)

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