$$ \newcommand{\cis}{\operatorname{cis}} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\paren}[1]{\left(#1\right)} \newcommand{\sq}[1]{\left[#1\right]} \newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ang}[1]{\left\langle#1\right\rangle} \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} \newcommand{\ceil}[1]{\left\lceil#1\right\rceil} \newcommand{\C}{\mathbb{C}} \newcommand{\D}{\mathbb{D}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\F}{\mathbb{F}} \newcommand{\T}{\mathbb{T}} \renewcommand{\S}{\mathbb{S}} \newcommand{\intr}{{\large\circ}} \newcommand{\limni}[1][n]{\lim_{#1\to\infty}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cG}{\mathcal{G}} \newcommand{\cH}{\mathcal{H}} \newcommand{\cI}{\mathcal{I}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\cK}{\mathcal{K}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cN}{\mathcal{N}} \newcommand{\cO}{\mathcal{O}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cS}{\mathcal{S}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cU}{\mathcal{U}} \newcommand{\cV}{\mathcal{V}} \newcommand{\cW}{\mathcal{W}} \newcommand{\cX}{\mathcal{X}} \newcommand{\cY}{\mathcal{Y}} \newcommand{\cZ}{\mathcal{Z}} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bI}{\mathbb{I}} \newcommand{\bJ}{\mathbb{J}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bL}{\mathbb{L}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bS}{\mathbb{S}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bU}{\mathbb{U}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bW}{\mathbb{W}} \newcommand{\bX}{\mathbb{X}} \newcommand{\bY}{\mathbb{Y}} \newcommand{\bZ}{\mathbb{Z}} $$
Due: March 19th, 2020

Math 185 Assignment 7

At the beginning of lecture on Tuesday, we will have the following definition:

Suppose that $\Omega \subseteq \C$ is open and $\gamma_0, \gamma_1 \subset \Omega$ are closed curves (though not necessarily from the same base point). A homotopy of (closed curves) $\gamma_0$ and $\gamma_1$ in $\Omega$ is a continuous function $h : [0, 1]\times[0, 1] \to \Omega$ so that for fixed $s$, $t\mapsto h(s, t)$ is a closed parameterized curve, and is a parameterization of $\gamma_0$ if $s = 0$ or $\gamma_1$ if $s = 1$.
Once again, this is a notion of $\gamma_0$ being able to be "smoothly deformed" into $\gamma_1$, but now we are allowed to move the endpoints of the curve.

  1. Homotopy is an equivalence relation

    Let $\Omega \subseteq \C$. Given closed curves $\gamma_0, \gamma_1$ in $\Omega$, let us write $\gamma_0 \sim_\Omega \gamma_1$ if they are homotopic in $\Omega$. Prove that $\sim_\Omega$ is an equivalence relation. That is, for any $\gamma_0, \gamma_1, \gamma_2$ closed curves in $\Omega$: $\gamma_0 \sim_\Omega \gamma_0$; $\gamma_0\sim_\Omega \gamma_1$ if and only if $\gamma_1\sim_\Omega\gamma_0$; and if $\gamma_0 \sim_\Omega \gamma_1$ and $\gamma_1\sim_\Omega\gamma_2$ then $\gamma_0\sim_\Omega\gamma_2$.

    (In fact, homotopy of (not necessarily closed) curves is also an equivalence relation, but showing this is essentially the same argument.)

  2. Invariance of integrals along homotopic closed curves

    Suppose that Ω is open, that $\gamma_0, \gamma_1$ are closed curves homotopic in $\Omega$, and $h : [0, 1]\times [0, 1] \to \Omega$ is a homotopy between them. Let $\sigma$ be the curve parameterized by \begin{align*} [0, 1] &\longrightarrow \Omega \\ s & \longmapsto h(s, 0) = h(s, 1). \end{align*}

    1. Show that $\gamma_0$ and $\sigma\,.\gamma_1\,.\sigma^-$ are homotopic as curves (rather than as closed curves). In particular, show that there is a homotopy between them which fixes the endpoints of the curves in question.
    2. Conclude that if $\gamma_0, \gamma_1$ are homotopic closed curves in $\Omega$ and $f : \Omega \to \C$ is holomorphic, then \[\int_{\gamma_0}f(z)\,dz = \int_{\gamma_1}f(z)\,dz.\]

  3. Homotopies in starlike regions

    Suppose that $\Omega \subseteq \C$ is open and starlike about $z$. Let $\gamma$ be a path in $\Omega$ from $w_0$ to $w_1$. Show that $\gamma$ is homotopic to $[w_0, z]\,.[z, w_1]$. Conclude that any two paths in $\Omega$ from $w_0$ to $w_1$ are homotopic to each other.

  4. Annuli

    Suppose $0 \leq r \lt R \leq \infty$ and $z_0 \in \C$. We denote by $A_{r, R}(z_0)$ the annulus of inner radius $r$ and outer radius $R$ centred at $z_0$: \[A_{r, R}(z_0) := \set{z \in \C \mid r \lt \abs{z-z_0} \lt R}.\]

    Suppose that $\Omega \subseteq \C$ is open, $0 \lt r \lt R \lt \infty$, $z_0 \in \C$, and $\overline{A_{r, R}(z_0)} \subset \Omega$.

    1. Show that $\partial B_{r}(z_0)$ and $\partial B_R(z_0)$ are homotopic in $\Omega$.
    2. Suppose that $f : \Omega \to \C$ is holomorphic. Show that for $z \in A_{r, R}$, we have \[f(z) = \frac1{2\pi i}\paren{\int_{\partial B_R(z_0)} \frac{f(w)}{w-z}\,dw - \int_{\partial B_r(z_0)} \frac{f(w)}{w-z}\,dw}.\]