Math 185 Assignment 9

1. Maximum modulus problems

   1. Suppose that $f, g : \overline{B_1(0)} \to \C$ are holomorphic on $B_1(0)$ and continuous, with $f(e^{it}) = g(e^{it})$ for all $t \in \R$. Show that $f \equiv g$.

   2. Suppose that $f : \overline{B_1(0)} \to \C$ is holomorphic on $B_1(0)$ and continuous, and that for some $\theta \gt 0$, $f(e^{it}) = 0$ for all $t \in [0, \theta]$. Show that $f \equiv 0$. (Hint: consider a product of functions of the form $z \mapsto f(\zeta z)$ with $|\zeta| = 1$.)

      Notice that this allows us to strengthen Part A: the result there still holds even if $f$ and $g$ agree merely on some arc of the circle.

2. Primitives on domains without simple connectivity

   Suppose that $\Omega \subseteq \C$ is a domain, but not necessarily simply connected, and that $f : \Omega \to \C$ is holomorphic. Show that $f$ admits a primitive on $\Omega$ if and only if for every closed curve $\gamma$ in $\Omega$, \[\int_\gamma f(w)\,dw = 0.\]

3. Some branches of logarithm

   1. Suppose that $\gamma$ is a curve in $\C$. Show that there is a unique unbounded connected component of $\C\setminus\gamma$. (Recall that for open sets $U \subseteq \Omega \subseteq \C$, $U$ is a connected component of $\Omega$ if and only if $U$ is non-empty and connected, and $U^c\cap \Omega$ is open.)

      It turns out to also be true that $\C\setminus\gamma$ has a unique bounded component whenever $\gamma$ is simple and closed, but this is much harder to show.

   2. Let $\alpha : [0, \infty) \to \C$ be continuous and injective (i.e., one-to-one), with $\alpha(0) = 0$ and $\lim_{t\to\infty}\abs{\alpha(t)} = \infty$. Set $V_\alpha = \C\setminus\alpha([0, \infty)).$

      Show that for any closed curve $\gamma$ in $V_\alpha$, \[\int_\gamma \frac{dz}z = 0.\] Conclude that $V_\alpha$ admits a branch of logarithm. (Hint: You may wish to apply homotopy invariance on $\C^\times$ rather than $V_\alpha$.)

      It turns out showing that $V_\alpha$ is connected is trickier than I realized. Show instead that the unbounded component $C$ of $V_\alpha$ admits a function $L : C \to \C$ such that $e^{L(z)} = z$ for all $z \in C$. As a bonus, show that $V_\alpha$ is connected and simply connected.

   3. Let $\alpha(t) = te^{it}$. For $z \in V_\alpha$, denote by $\theta_z$ the unique element of $(0, 2\pi)$ for which $|z|e^{i|z|}=ze^{i\theta_z}$. Compute, in terms of $\theta_z$, an explicit formula for the branch of logarithm $\log_{V_\alpha}$ (i.e., the one defined on $V_\alpha$ which takes the value $0$ at $1$.)

      Draw a sketch of the region $\log_{V_\alpha}(V_\alpha)$, but do not submit it.

4. Some exponentiation

   In the following question, we will work always with the principal branch of $z^\alpha$: the one defined for $z \in \C\setminus(-\infty, 0]$ corresponding to the logarithm taking value $0$ at $1$.
   1. Compute $(-i)^{1/2}$ and $5^{i+1}$.
   2. Let $\alpha \gt 0$. Determine all points $t_0 \in (-\infty, 0]$ for which \[\lim_{\substack{z \to t_0 \\ z \in \C\setminus(-\infty, 0]}} z^\alpha\] exists.
   3. Determine the range of $z^i$. (I recommend, but do not require, that you also produce a diagram indicating the value $z^i$ takes on a typical ray from the origin and on a typical circle about $0$.)

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.

• Another zero counting problem

  Let $\lambda \gt 1$, and define $f_\lambda(z) = \lambda - z - e^{-z}$. Prove that $f_\lambda$ has a unique zero in the right half-plane $H = \set{z \in \C \mid \Re(z) \gt 0}$. Moreover, if $x_\lambda$ is this zero, show that $x_\lambda \in \R$ and $\lim_{\lambda\to1^+}x_\lambda = 0$.