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Due: February 20th, 2020

Math 185 Assignment 4

  1. Discs in open sets

    Let $\Omega \subseteq \C$ be open, and suppose $z \in \Omega$, $r \gt 0$ are such that $\partial B_r(z) \subset \Omega$ (that is, the circle of radius $r$ centred at $z$ is contained in $\Omega$). Show that there is some $R \gt r$ so that $B_R(z) \subseteq \Omega$. (Hint: $\partial B_r(z)$ is compact.)

    Correction: the assumption should be that $\overline{B_r(z)} = \partial B_r(z) \cup B_r(z) \subset \Omega$; otherwise there are many counterexamples.

  2. Starlike regions

    Recall that a set $V \subseteq \C$ (or $\R^n$) is called starlike about $z_0$ if for any $z \in V$, we have $[z_0, z] \subseteq V$. A set is starlike if it is starlike about some point it contains.

    Recall that we proved the following in class:

    Suppose that $\Omega \subseteq \C$ is non-empty, open, and convex, and $f : \Omega \to \C$ is a continuous function such that for any triangle $T \subset \Omega$, \[\int_{\partial T} f(z)\,dz = 0.\] Then $f$ admits a primitive on $\Omega$.
    Show that the same conclusion is true if we merely assume that $\Omega$ is non-empty, open, and starlike. (It is enough to identify which part(s) of the proof used convexity, and show that they still work; you do not need to repeat parts of the argument that require no update.)

  3. Convex sets

    1. Show that for any $z_0 \in \C$ and any $r \gt 0$, the set $B_r(z_0) = \set{z \in \C \mid \abs{z-z_0} \lt r}$ is convex.
    2. Show that a set $V \subseteq \C$ is convex if and only if for every $n \in \N$, $z_1, \ldots, z_n \in V$, and $a_1, \ldots, a_n \in [0, 1]$ with $a_1+\cdots+a_n = 1$, we have \[a_1z_1 + a_2z_2 + \cdots + a_nz_n \in V.\]

  4. Radii of convergence

    1. Suppose that $\Omega \subseteq \C$, $f : \Omega \to \C$ is holomorphic, $z_0 \in \Omega$, and $r \gt 0$ is such that $\overline{B_r(z_0)} \subset \Omega$. Recall that on the disc $B_r(z_0)$, $f$ is given by the power series \[f(z) = \frac1{2\pi i} \sum_{n=0}^\infty \paren{\int_{\partial B_r(z_0)} \frac{f(w)}{(w-z_0)^{n+1}}\,dw}(z-z_0)^n.\] Show that radius of convergence of the series is at least \[\sup\set{R \geq 0 \mid \overline{B_R(z_0)} \subset \Omega}.\]
    2. Let $f : \R \to \R$ be given by \[f(t) = \frac1{1+t^2}.\] Show that $f$ is $\R$-analytic, in the sense that for every $t_0 \in \R$ there are coefficients $a_n \in \R$ and some $\delta \gt 0$ so that for all $t \in (t_0-\delta, t_0+\delta)$, \[f(t) = \sum_{n=0}^\infty a_n(t-t_0)^n.\] Calculate, for every $t_0$, the radius of convergence of this series.

  5. Some additional integrals for your enjoyment

    Evaluate the following integrals.
    1. \[\int_{\partial B_r(0)} \frac{\sin(z)}z\,dz \quad(r \gt 0)\quad\]
    2. \[\int_{\partial B_2(0)} \frac{\cos(z)}{1+z^2}\,dz\]
    3. \[\int_{\partial B_r(0)} \frac{z^2+1}{z(z^2+4)}\,dz \quad(r \gt 2 \text{ and } 2 \gt r \gt 0)\]
    Feel free to invoke all the techniques we have available, including Cauchy's integral formula and algebraic manipulations such as partial fractions.

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.