Additional Problems for Chapter 10 \( \newcommand{\ds}{\displaystyle} \newcommand{\diam}{\operatorname{diam}} \newcommand{\area}{\operatorname{area}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\res}{\operatorname{res}} \newcommand{\wind}{\operatorname{W}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\iso}{\stackrel{\cong}{\longrightarrow}} \newcommand{\myint}{\operatorname{int}} \newcommand{\ve}{\varepsilon} \newcommand{\es}{\emptyset} \newcommand{\sm}{\smallsetminus} \newcommand{\bd}{\partial} \newcommand{\Chat}{\hat{\mathbb C}} \newcommand{\Cstar}{{\mathbb C}^*} \newcommand{\Dstar}{{\mathbb D}^*} \newcommand{\myre}{\operatorname{Re}} \newcommand{\myim}{\operatorname{Im}} \newcommand{\ov}{\overline} \newcommand{\io}{\iota} \newcommand{\con}{\operatorname{const.}} \newcommand{\OO}{{\mathcal O}} \newcommand{\MM}{{\mathcal M}} \newcommand{\FF}{{\mathcal F}} \newcommand{\CC}{{\mathbb C}} \newcommand{\PP}{{\mathbb P}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HH}{{\mathbb H}} \newcommand{\TT}{{\mathbb T}} \newcommand{\II}{{\mathbb I}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\DD}{{\mathbb D}} \newcommand{\QQ}{{\mathbb Q}} \)




Additional Problems for Chapter 10

  1. Let $\{ c_n \}$ be a sequence of non-zero complex numbers such that $\sum_{n=1}^{\infty} |c_n|$ converges, and let the sequence $\{ z_n \}$ form a dense subset of the unit circle $\TT$. Show that \[ f(z) = \sum_{n=1}^{\infty} \frac{c_n}{z-z_n} \] defines a holomorphic function in $\DD$, with $\TT$ as its natural boundary.
  2. Suppose $f \in \OO(\CC)$ takes on real values on both the real and imaginary axes. Show that $f(z)=g(z^2)$ for some $g \in \OO(\CC)$.
  3. Let $f(z)=\sum_{n=0}^{\infty} a_n \, z^n$ be an entire function. Under what condition on $\{ a_n \}$ does $f$ map the real axis to the imaginary axis? Under what condition on $\{ a_n \}$ is $f$ one-to-one?
  4. Suppose $f \in \OO(\Cstar)$ has a simple pole at $0$ and $f(\TT) \subset \RR$. Show that \[ f(z)=\frac{a}{z}+b+ \bar{a} \, z \] for some constants $a \in \Cstar$ and $b \in \RR$.
  5. What can you say about a bounded holomorphic function in the domain $\{ z \in \CC: |z-i|>1/2 \}$ which takes real values on the segment $[-1,1] \subset \RR$?
  6. Suppose $f$ is bounded and holomorphic in the vertical strip $\{ z \in \CC : 0< \myre(z) < 1 \}$ and $\myim(f(z)) \to 0$ as $z$ tends to any point on the boundary lines $\myre(z)=0$ or $\myre(z)=1$. Show that $f$ is constant.
  7. Let $f=u+iv$ be holomorphic in $U = \{ z \in \DD: \myim(z)>0 \}$ and continuous on $\ov{U}$. Suppose $u=0$ on $[0,1] \subset \bd U$ and $v=0$ on $[-1,0] \subset \bd U$. Show that there is a constant $C>0$ such that $|f(z)| \leq C |z|^{1/2}$ for all $z \in U$.
  8. Show that $\zeta: \HH \to \HH$ is an anti-holomorphic involution if and only if \[ \zeta(z) = \frac{a \ov{z}+b}{c \ov{z}-a}, \] with $a,b,c \in \RR$ and $a^2+bc=1$.
  9. Let $U \subset \CC$ be simply connected and $a_0, \cdots, a_{n-1} \in \OO(U)$. Assume that for each $t \in U$ the polynomial equation \[ P(z,t) = \sum_{k=0}^{n-1} a_k(t)z^k + z^n = 0 \] has $n$ distinct solutions in $z$. Prove that there exist $f_1, \cdots, f_n \in \OO(U)$ such that $P(f_k(t),t)=0$ for all $t \in U$ and $1 \leq k \leq n$. (Hint: Solve the problem locally on each small disk in $U$, then use the monodromy theorem.)

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