\[ \newcommand{\Arg}{\mathsf{Arg}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Im}{\mathsf{Im}} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\ball}{\mathsf{B}} \newcommand{\wind}{\mathsf{wind}} \]

Week 3 Worksheet

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Power Series

  1. Determine the radius of convergence of each of these power series.
    1. $\displaystyle \sum_{n=0}^\infty \frac{2^n}{n} z^n$
    2. $\displaystyle \sum_{n=0}^\infty n! z^n$
    3. $\displaystyle \sum_{n=0}^\infty n^p z^n$ for a fixed $p \in \N$.
  2. The zeroth Bessel function $J_0(z)$ is defined by \[ J_0(z) = \sum_{n=0}^\infty (-1)^n \frac{1}{(n!)^2} \frac{z^n}{2n} \] but what is its radius of convergence?
  3. The power series \[ \sum_{n=0}^\infty a_n z^n \] and \[ \sum_{n=0}^\infty b_n z^n \] have radii of convergence $R > 0$ and $S > 0$ respectively. What is the radius of convergence of the series $\displaystyle \sum_{n=0}^\infty (a_n + b_n) z^n$?

Differentiation of Power Series

  1. Fix $A$ and $B$ distinct complex numbers. Prove that \[ \dfrac{A^n - B^n}{A - B} = A^{n-1} B^0 + A^{n-2} B^1 + \cdots + A^1 B^{n-2} + A^0 B^{n-1} \] for all $n \in \N$.
  2. Starting from the geometric series \[ \dfrac{1}{1-z} = 1 + z + z^2 + z^3 + \cdots \] on $\ball(0,1)$ find a power-series representation for each of the following functions on $\ball(0,1)$.
    1. $f(z) = \dfrac{1}{(1-z)^2}$
    2. $g(z) = \dfrac{1}{1-z^2}$
    3. $h(z) = \dfrac{2z}{(1-z^2)^2}$

Special Functions

  1. Prove that $\{ \exp(z) : z \in \C \} = \C \setminus \{0\}$.
  2. A number $\tau$ is a \define{period} of a function $f : \C \to \C$ if $f(z + \tau) = f(z)$ for all $z \in \C$.
    1. Verify that $2 \pi i n$ is a period of $\exp$ for every $n \in \Z$.
    2. Does $\exp$ have any other periods?
  3. Determine the real and imaginary parts of $\exp$ and verify the Cauchy-Riemann equations are satisfied everywhere.
  4. For which complex numbers $z$ is $\exp(z)$ real? Complex?
  5. Find the zeroes of $f(z) = 1 + \exp(z)$ and $g(z) = 1+i - \exp(z)$.
  6. Evaluate $\cos(i)$ and $\sin(i)$.
  7. Can the ratio test be applied to the power series defining $\cos$ and $\sin$?
  8. Verify the addition formulae \[ \begin{aligned} \cos(z+w) &= \cos(z) \cos(w) - \sin(z) \sin(w) \\ \sin(z+w) &= \sin(z) \cos(w) + \sin(w) \cos(z) \end{aligned} \] for all $z,w \in \C$.
  9. Verify that $\exp(z) = \cosh(z) + \sinh(z)$ for all $z \in \C$.
  10. Verify that $(\cosh(z))^2 - (\sinh(z))^2 = 1$ for all $z \in \C$.