Math 185 Assignment 7

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

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 \[f(z) = \frac1{2\pi i}\paren{\int_{\partial B_R(z_0)} \frac{f(w)}{w-z}\,dw - \int_{\partial B_r(z_0)} \frac{f(w)}{w-z}\,dw}.\]