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Due: January 30th, 2020

Math 185 Assignment 1

  1. Some computations

    1. Find $x, y \in \R$ so that $x + iy$ is a primitive 16th root of unity. Express your answer using rationals and radicals. There are multiple correct answers to this problem, as there is more than one primitive 16th root of unity. You may without proof any of the identities and values of trigonometric functions present on the linked Wikipedia page List of trigonometic functions (as of 21:21, 15 January 2020). You will notice that this includes the values of $\sin$ and $\cos$ on angles of the form $\frac{k\pi}{n}$ with $n \in \set{2, 3, 4, 6}$ but not larger, so some computation will be necessary.
    2. Prove that if $a \in \C\setminus\set0$, the equation $z^n=a$ has exactly $n$ distinct solutions in $\C$. (You may find it useful that $\sin$ and $\cos$ are both $2\pi$-periodic.)
    3. Compute all solutions of the equation $z^6 = -\frac{27}2 + i\frac{27\sqrt3}2$. Express your answer(s) in the form $r\cis(\theta)$.
    4. Let $n \gt 1$ and let $\zeta \neq 1$ be an $n$-th root of unity. Prove that \[\sum_{k=0}^{n-1}\zeta^k = 0.\]

  2. Some useful facts about holomorphic functions

    Let $\Omega \subseteq \C$ be open, and $z \in \Omega$.
    1. Suppose that $f : \Omega \to \C$ is holomorphic at $z$. Prove that $f$ is continuous at $z$.
    2. Prove the product rule: show that if $f, g : \Omega \to \C$ are holomorphic at $z$, then $fg$ is holomorphic at $z$ with \[(fg)'(z) = f'(z)g(z) + f(z)g'(z).\]

  3. Some counterexamples

    Give examples of functions $f = u + iv$ defined on $\C$ so that:
    1. $f$ is holomorphic at $0$ but no other point in $\C$;
    2. $f$ satisfies the Cauchy-Riemann equations at $0$ and is continuous at $0$, but is not holomorphic at $0$;
    3. $f$ is holomorphic at $0$, the partial derivatives of $u$ with respect to $x$ and $y$ exist in a neighbourhood of $0$, but at least one of those partial derivatives fails to be continuous at $(0,0)$. (A set $S \subseteq \C$ is called a "neighbourhood of $z_0$" if $S$ is open and $z_0 \in S$.)
    (You may find the function $g : \R \to \R$ given by \[g(t) = \begin{cases}0 & \text{ if } t = 0 \\ t^2\sin\paren{\frac1t} & \text{ otherwise}\end{cases}\] to be useful.)

  4. A fact that will be useful for power series

    Recall that for a sequence $(a_n)_n$ in $\R$, its limit superior is defined by \[\limsup_{n\to\infty} \, a_n := \lim_{n \to \infty} \paren{\sup\set{a_k \mid k \gt n}} \in \R\cup\set{\pm\infty}.\] The limit superior of a sequence always exists, and—if the sequence is bounded—is finite. Before proceeding to the following problem, spend some time to convince yourself that $\displaystyle\limsup_{n\to\infty} \, a_n$ is the unique $x \in \R\cup\set{\pm\infty}$ with the following two properties:

    • if $y \gt x$, then $a_n \gt y$ for only finitely many $n$; and
    • if $y \lt x$, then $a_n \gt y$ infinitely often.

    Suppose that $(c_n)_n$ is a sequence in $\C$, and define \[R = \sup\set{r \geq 0 \mid (r^n c_n)_n \text{ is bounded}}.\]

    Show that if $R \gt 0$ then \[\limsup_{n\to\infty} \abs{c_n}^{1/n} = 1/R,\] interpreting $1/\infty$ as $0$. Also show that if $R = 0$ if and only if the sequence $\paren{\abs{c_n}^{1/n}}_n$ is unbounded.

    You may be cavalier with the use of standard limits such as \[\limni A^{\frac1n} = 1 \qquad\text{ for } A \in (0, \infty),\] if you find them helpful.

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.