\[ \newcommand{\Ann}{\mathsf{Ann}} \newcommand{\Arg}{\mathsf{Arg}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Im}{\mathsf{Im}} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\ball}{\mathsf{B}} \newcommand{\wind}{\mathsf{wind}} \newcommand{\Log}{\mathsf{Log}} \newcommand{\l}{<} \]

Week 9 Worksheet

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Laurent Series

  1. For each of the following functions determine its Laurent series on the given annulus.
    1. $\dfrac{1}{z-3}$ on $\Ann(0,3,\infty)$
    2. $\dfrac{1}{z(1-z)}$ on $\Ann(0,0,1)$
    3. $\cos(1/z)\vphantom{\dfrac{1}{z}}$ on $\Ann(0,0,\infty)$
  2. Define $f : \C \setminus \{-1,3\}$ by $f(z) = \dfrac{1}{z+1} + \dfrac{1}{z-3}$. Calculate its Laurent series on $\Ann(0,0,1)$, $\Ann(0,1,3)$ and $\Ann(0,3,\infty)$.
  3. Define $f : \C \setminus \{0,1\} \to \C$ by $f(z) = \dfrac{1}{z^2(z-1)}$.
    1. Calculate its Laurent series on $\Ann(0,0,1)$.
    2. Calculate its Laurent series on $\Ann(1,0,1)$.
  4. Define $f : \C \setminus \{1\} \to \C$ by $f(z) = \dfrac{1}{(z-1)^2}$.
    1. Calculate its Laurent series on $\Ann(1,0,\infty)$.
    2. Calculate its Laurent series on $\Ann(0,0,1)$.
    3. Calculate its Laurent series on $\Ann(0,1,\infty)$.

Isolated Singularities

  1. For each of the following functions determine all the poles and their orders.
    1. $f(z) = \dfrac{1}{z^2 + 1}$
    2. $f(z) = \dfrac{1}{z^4 + 16}$
    3. $f(z) = \dfrac{1}{z^4 + 2z^2 + 1}$
    4. $f(z) = \dfrac{1}{z^2 + z - 1}$
  2. What type of singularity does each of the following functions have at the origin?
    1. $f(z) = \sin(1/z)$
    2. $f(z) = \dfrac{1}{z^3} (\sin(z))^2$
    3. $f(z) = \dfrac{\cos(z) - 1}{z^2}$
  3. Fix a domain $D$ and $b \in D$. Suppose $f$ is holomorphic and bounded on $D \setminus \{b\}$. Prove that $f$ has a removable singularity at $b$.