Additional Problems for Chapter 2 \( \newcommand{\ds}{\displaystyle} \newcommand{\diam}{\operatorname{diam}} \newcommand{\area}{\operatorname{area}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\res}{\operatorname{res}} \newcommand{\wind}{\operatorname{W}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\iso}{\stackrel{\cong}{\longrightarrow}} \newcommand{\myint}{\operatorname{int}} \newcommand{\ve}{\varepsilon} \newcommand{\es}{\emptyset} \newcommand{\sm}{\smallsetminus} \newcommand{\bd}{\partial} \newcommand{\Chat}{\hat{\mathbb C}} \newcommand{\Cstar}{{\mathbb C}^*} \newcommand{\Dstar}{{\mathbb D}^*} \newcommand{\myre}{\operatorname{Re}} \newcommand{\myim}{\operatorname{Im}} \newcommand{\ov}{\overline} \newcommand{\io}{\iota} \newcommand{\con}{\operatorname{const.}} \newcommand{\OO}{{\mathcal O}} \newcommand{\MM}{{\mathcal M}} \newcommand{\FF}{{\mathcal F}} \newcommand{\CC}{{\mathbb C}} \newcommand{\PP}{{\mathbb P}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HH}{{\mathbb H}} \newcommand{\TT}{{\mathbb T}} \newcommand{\II}{{\mathbb I}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\DD}{{\mathbb D}} \newcommand{\QQ}{{\mathbb Q}} \)




Additional Problems for Chapter 2

  1. Sketch the images of the closed curves $\gamma, \eta: [0,2\pi] \to \CC$ defined by \[ \gamma(t) = \sin t \ \, e^{it} \qquad \eta(t) = \sin t \ \, e^{2it}. \] Find the values of the winding numbers $\wind(\gamma,\cdot)$ and $\wind(\eta,\cdot)$ in each connected component of $\CC \sm |\gamma|$ and $\CC \sm |\eta|$, respectively.
  2. Let $f$ be an entire function with $f^{-1}(0)=\{ 0 \}$ and $f'(0)\neq 0$. Show that there is entire function $g$ which satisfies $(g(z))^2=f(z^2)$ for all $z \in \CC$. Verify that any such $g$ is an odd function, i.e., $g(-z)=-g(z)$ for all $z$.
  3. Suppose $f,g \in \OO(\CC)$ satisfy $f^2+g^2=1$ everywhere in $\CC$. Prove that $f=\cos(h)$ and $g=\sin(h)$ for some $h \in \OO(\CC)$. (Hint: $f+ig \in \OO(\CC)$ is nowhere vanishing.)
  4. Describe all $f \in \OO(\CC)$ for which $f^2+(f')^2$ is a constant function.
  5. Let $f$ denote the primitive of $1/(1+z^2)$ in the doubly slit plane $U := \CC \sm \{ \pm iy: y \geq 1 \}$ which satisfies $f(0)=0$. Show that $\tan(f(z))=z$ for all $z \in U$, where as usual $\tan$ is the meromorphic function defined by $\tan z=\sin z / \cos z$. (Hint: First show that $f(\tan z)=z$ for all $z$ in some neighborhood of $0$.)
  6. Suppose $f: \CC \to \CC$ is a function whose powers $f^2$ and $f^3$ are polynomials. Show that $f$ itself must be a polynomial. (Hint: The polynomials $P:=f^2, Q:=f^3$ satisfy $P^3=Q^2$. Find an entire function which is simultaneously a holomorphic square root of $P$ and a holomorphic cube root of $Q$. Notice that if $f$ were assumed continuous to begin with, any single integer power of $f$ being a polynomial would be enough to guarantee $f$ is a polynomial.)
  7. Let $p \in \DD$ and $\varphi(z)=(z-p)/(1-\ov{p}z)$. Show that \[ \frac{1}{\pi}\iint_{\DD} |\varphi'(z)| dx dy = \frac{1-|p|^2}{|p|^2} \, \log \frac{1}{1-|p|^2}. \] (Hint: Switch to polar coordinates $r,\theta$. For each $r$ express the integral over $\theta$ as a complex integral which can be evaluated using Cauchy's integral formula.)

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