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Due: April 2nd, 2020

Math 185 Assignment 8

$$\newcommand{\res}{\operatorname{res}}$$

  1. More residues

    Suppose that $f : D_r'(z_0) \to \C$ is holomorphic, with a simple pole at $z_0$. Let $g : B_r(z_0)$ be holomorphic with $g(z)f(z) = 1$ for $z$ in the domain of $f$. Show that \[\res_{z_0}(f) = \frac{1}{g'(z_0)}.\] (Notice, also, that this approach will fail if $f$ has a pole of order more than one at $z_0$.)

  2. Some integrals

    1. Let $p, q$ be polynomials with complex coefficients, so that $q$ admits no roots in $\R$ and $\deg q \geq \deg p + 2$. Show that \[\int_{-\infty}^\infty \frac{p(x)}{q(x)}\,dx = 2\pi i \sum_{j=1}^k \res_{z_j}\paren{\frac pq},\] where $z_1, \ldots, z_k$ are the zeros of $q$ with $\Im(z_k) \gt 0$.
    2. Compute, for $n \in \N$, $n \geq 2$, the integral \[\int_0^\infty \frac{dx}{x^n+1}.\] (Hint: use the "wedge" $[0, R]\,. \sigma_R\,. [R\zeta_{n}, 0]$ where $\zeta_n = e^{2\pi i / n}$ and $\sigma_R(t) = Re^{it}$ for $t \in [0, 2\pi/n]$.)
    3. Compute the integral \[\int_{-\infty}^\infty \frac{2x^2+1}{(x^4+1)(x^2+1)}\,dx.\]

  3. Perturbations of zeros

    Suppose that $r \gt 1$, and $f, g : B_{r}(0) \to \C$ are holomorphic. Suppose further that $f$ has a simple zero at $0$, and vanishes nowhere else in $\overline{B_1(0)}$. Let $f_\epsilon = f(z) + \epsilon g(z)$.

    1. Show that if $\epsilon$ is sufficiently small, then $f_\epsilon$ has a unique zero in $B_1(0)$.
    2. Let $z_\epsilon$ be the unique zero of $f_\epsilon$ from part A. Show that $\epsilon \mapsto z_\epsilon$ is continuous.

    (A consequence of this is that the zeros of a polynomial are continuous functions of its coordinates. Reflect on how to state this precisely.)

  4. A counting problem

    Determine the number of zeros of $p(z) = 1 + 6z^3 + 3z^{10}+z^{11}$ in $A_{1, 2}(0)$. (Hint: determine the number of zeros in $B_1(0)$ and in $B_2(0)$. This can be done using either the Argument Principle or Rouché's Theorem and some estimates.)

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.