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Due: March 12th, 2020

Math 185 Assignment 6

  1. Essential singularitites

    Suppose that $\Omega \subset \C$ is open, and $f : \Omega \to \C$ is holomorphic with an essential singularity at $z_0$.

    1. Suppose $g : \Omega \cup\set{z_0} \to \C$ is holomorphic with a zero of order $N$ at $z_0$. Classify the singularity of $fg$ at $z_0$.
    2. Suppose $g : \Omega \to \C$ has a pole at $z_0$. Classify the singularity of $fg$ at $z_0$.

  2. Dealing with an essential singularity

    Let $\Omega = \C\setminus\set0$, and consider the function $f : z \mapsto \exp\paren{\frac1z}$ which is holomorphic on $\Omega$.

    1. Show that there is are $a_0, a_{-1}, a_{-2}, \ldots, \in \C$ so that the series \[\sum_{n=-\infty}^0 a_nz^n\] converges uniformly to $f$ on compact subsets of $\Omega$.
    2. Find $a_{-1}$, the residue of $f$ at $0$, explicitly.
    3. Compute \[\int_{\partial \color{red}{B_{123}(0)}} f(z)\,dz.\]

  3. No poles of order 1/2

    Suppose $f : D'_r(z_0) \to \C$ is holomorphic, and that for some $\epsilon \gt 0$ and $A \in \R$ we have \[\abs{f(z)} \leq A\abs{z-z_0}^{\epsilon - 1}.\] Show that $f$ has a removable singularity at $0$.

  4. The Riemann sphere

    Recall that we defined the Riemann sphere as \[\S := \set{(\alpha, \beta, \gamma) \in \R^3 \mid \alpha^2+\beta^2+\gamma^2=1}.\] We identified $\S$ with the "infinitied" complex plane $\C_\infty$ by \begin{align*} S &\longrightarrow \C_\infty\\ (\alpha, \beta, \gamma) &\longmapsto \begin{cases} \infty & \text{ if } \gamma = 1 \\ \frac{\alpha+i\beta}{1-\gamma} & \text{ otherwise.}\end{cases} \end{align*} Here the inverse mapping is given by \[ x+iy \longmapsto \paren{\frac{2x}{x^2+y^2+1}, \frac{2y}{x^2+y^2+1}, 1-\frac{2}{x^2+y^2+1}}. \]

    1. Describe the map $\S \to \S$ which corresponds to $z \mapsto iz$.
    2. Describe the map $\S \to \S$ which corresponds to $z \mapsto \bar z$.
    3. Describe the map $\C_\infty \to \C_\infty$ which corresponds to sending each point $p \in \S$ to its antipode, $-p$.

    (An example of the level of detail desired: the map $z \mapsto z^{-1}$ corresponds to a rotation by angle $\pi$ about the line through $1$ and $-1$ (i.e., through $(1, 0, 0)$ and $(-1, 0, 0)$). Of course, some justification of this claim would be required.)

  5. Meromorphisms and poles

    For each of the following functions, find the largest domain on which they are meromorphic and classify their singularities.

    1. \[f(z) = e^z + e^{-\frac1z}\]
    2. \[g(z) = \frac{1}{\sin(1/z)}\]