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Due: April 23rd, 2020

Math 185 Assignment 11

$$ \newcommand{\res}{\operatorname{res}} \newcommand{\H}{\mathbb{H}} $$

  1. Functions with non-zero derivative

    1. Give an example of open sets $U, V \subseteq \C$ and a function $f : U \to V$ so that $f'(z) \neq 0$ for all $z \in U$, $f(U) = V$, but $f$ is not injective.

    2. Suppose $U, V \subseteq \C$ are open, and $f : U \to V$ is holomorphic. We say $f$ is a local bijection if for every $z_0 \in U$ there is $\delta \gt 0$ so that \[f|_{B_\delta(z_0)} : B_\delta(z_0) \longrightarrow f(B_\delta(z_0))\] is a bijection.

      Prove that $f$ is a local bijection if and only if $f'(z) \neq 0$ for all $z \in U$.

  2. Boundary values on the upper half-plane

    Suppose $f : \H \to \C$ is holomorphic and extends continuously to $\H \cup \R \cup \set{\infty}$. Show that if for all $x$ in some interval $[a, b] \subseteq \R$ (with $a \lt b$) we have $f(x) = 0$ then $f \equiv 0$. (Hint: you may find a result from a previous assignment to be useful here.)

    As a bonus, determine if the same holds if we merely assume that $f$ extends continuously to $\H \cup [a, b]$, not necessarily all of $\H\cup\R\cup\set{\infty}$.

  3. Simple connectivity

    Suppose that $U, V \subseteq \C$ are conformally equivalent. Prove that $U$ is simply connected if and only if $V$ is.

  4. The plane, the disc, and the sphere are distinct

    Recall that a function $f$ defined on $\C_\infty$ is holomorphic if it is holomorphic at every point in $\C$ and has a removable singularity at $\infty$. Prove that no two of $\C$, $\C_\infty$, and $\D$ are conformally equivalent.