\[
\newcommand{\Arg}{\mathsf{Arg}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Im}{\mathsf{Im}}
\newcommand{\intd}{\,\mathsf{d}}
\newcommand{\Re}{\mathsf{Re}}
\newcommand{\ball}{\mathsf{B}}
\newcommand{\wind}{\mathsf{wind}}
\]
Week 4 Worksheet
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Differentiation of Power Series
- 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 $n \in \N$.
- 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 $\ball(0,1)$.
- $f(z) = \dfrac{1}{(1-z)^2}$
- $g(z) = \dfrac{1}{1-z^2}$
- $h(z) = \dfrac{2z}{(1-z^2)^2}$
The Exponential Function and Trigonometric Functions
- Prove that $\{ \exp(z) : z \in \C \} = \C \setminus \{0\}$.
- A number $\tau$ is a \define{period} of a function $f : \C \to \C$ if $f(z + \tau) = f(z)$ for all $z \in \C$.
- Verify that $2 \pi i n$ is a period of $\exp$ for every $n \in \Z$.
- Does $\exp$ have any other periods?
- Determine the real and imaginary parts of $\exp$ and verify the Cauchy-Riemann equations are satisfied everywhere.
- For which complex numbers $z$ is $\exp(z)$ real? Complex?
- Find the zeroes of $f(z) = 1 + \exp(z)$ and $g(z) = 1+i - \exp(z)$.
- Evaluate $\cos(i)$ and $\sin(i)$.
- Can the ratio test be applied to the power series defining $\cos$ and $\sin$?
- 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 $z,w \in \C$.
- Verify that $\exp(z) = \cosh(z) + \sinh(z)$ for all $z \in \C$.
- Verify that $(\cosh(z))^2 - (\sinh(z))^2 = 1$ for all $z \in \C$.
Contours
- Compute the length of the smooth path $\gamma(t) = t^3 + i t$ on $[3,5]$.
- Let $R$ be the rectangle with vertices $0$, $2$, $2+3i$ and $3i$. Give formulas for smooth paths $\gamma_1$, $\gamma_2$, $\gamma_3$, $\gamma_4$ such that the contour $\Gamma = (\gamma_1,\gamma_2,\gamma_3,\gamma_4)$ traverses the perimeter of $R$ once anti-clockwise.
- The \define{reverse} of a path $\gamma : [a,b] \to \C$ is the path $\tilde{\gamma} : [a,b] \to \C$ given by $\tilde{\gamma}(t) = \gamma(a+b-t)$.
- Draw the reverse of $\gamma(t) = it + 2(1-t)$ on $[0,1]$.
- How would you define the reverse of a contour $\Gamma = (\gamma_1,\gamma_2,\gamma_3)$?