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

Week 10 Worksheet

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Poles and Residues

  1. For each of the following functions determine its poles, their orders, and their residues.
    1. $\dfrac{1}{z(1-z^2)}$ on $\C \setminus \{0,1,-1\}$
    2. $\dfrac{\sin(z)}{\cos(z)}$ on $\C \setminus (\pi \Z + \frac{\pi}{2})$
    3. $\dfrac{z}{1+z^4}$ on $\C \setminus \{e^{i \pi / 4}, e^{3i \pi / 4}, e^{5 i \pi / 4}, e^{7 i \pi / 4} \}$
    4. $\left( \dfrac{z+1}{z^2 + 1} \right)^2$ on $\C \setminus \{i,-i\}$
  2. Determine the singularities of the following functions and use Taylor series to calculate their residues.
    1. $\dfrac{\sin(z)}{z^2}$
    2. $\dfrac{(\sin z)^2}{z^4}$
  3. Define $f : \C \setminus \{0,1\} \to \C$ by $f(z) = \dfrac{1}{z (1-z)^2}$. It has singularities at $0$ and $1$.
    1. Determine the Laurent series of $f$ on $\Ann(0,0,1)$ and $\Ann(1,0,1)$.
    2. Use your answers to write down the orders of any poles and the residues of the singularities.
    3. Verify your answers to b) using the lemmas from the video "Calculating Residues".
  4. Define \[ f(z) = \left( \dfrac{z+1}{z-1} \right)^3 \] on $\C \setminus \{1\}$. Calculate $\Res(f,1)$ by determining the Laurent series of $f$ on $\Ann(1,0,\infty)$.
  5. Determine the residue of $\dfrac{1}{z^2 \sin(z)}$ at zero.
  6. Suppose $f,g : D \to \C$ are holomorphic and $b \in D$. Assuming $f$ has a zero of order $m$ at $b$ and $g$ has a zero of order $n$ at $b$ prove that $f(z) g(z)$ has a zero of order $n+m$ at $b$.

Cauchy's Residue Theorem

  1. Given $r > 0$ write $\gamma_r(t) = re^{it}$ on $[0,2\pi]$. Use Cauchy's residue theorem to evaluate each of the following integrals.
    1. $\displaystyle\int\limits_{\gamma_4} \dfrac{1}{z^2 - 5z + 6} \intd z$
    2. $\displaystyle\int\limits_{\gamma_{5/2}} \dfrac{1}{z^2 - 5z + 6} \intd z$
    3. $\displaystyle\int\limits_{\gamma_2} \dfrac{\exp(az)}{1 + z^2} \intd z$ where $a \in \R$