\[
\newcommand{\Arg}{\mathsf{Arg}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Im}{\mathsf{Im}}
\newcommand{\intd}{\,\mathsf{d}}
\newcommand{\Re}{\mathsf{Re}}
\newcommand{\ball}{\mathsf{B}}
\newcommand{\wind}{\mathsf{wind}}
\]
Week 3 Worksheet
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Power Series
- Determine the radius of convergence of each of these power series.
- $\displaystyle \sum_{n=0}^\infty \frac{2^n}{n} z^n$
- $\displaystyle \sum_{n=0}^\infty n! z^n$
- $\displaystyle \sum_{n=0}^\infty n^p z^n$ for a fixed $p \in \N$.
- 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?
- 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 $\displaystyle \sum_{n=0}^\infty (a_n + b_n) z^n$?
Holomorphic Functions
- Use the definition of the derivative to differentiate each of the following functions.
- $f(z) = z^2 + z$
- $f(z) = \dfrac{1}{z}$
- Define $f : \C \to \C$ by $f(z) = |z|^2$.
- Verify that $f$ is differentiable at the origin.
- Is $f$ holomorphic on any domain?
Cauchy-Riemann Equations
- If $h : \R^2 \to \R$ is defined by $h(x,y) = 2xy$ use the definitions of the partial derivatives to calculate $\partial_1 h$ and $\partial_2 h$.
- Verify that the real and imaginary parts of $f(z) = 1/z$ together satisfy the Cauchy-Riemann equations.
- Use the Cauchy-Riemann equations to determine whether there are any $z \in \C$ where $f(z) = |z|$ is differentiable.
- Verify that the functions
\[
u(x,y) = x^3-3xy^2 \qquad v(x,y) = 3x^2y-y^3
\]
satisfy the Cauchy-Riemann equations for all $(x,y)$. Show that $u, v$ are the real and imaginary parts of a holomorphic function $f : \C \to \C$.
- Verify that the functions
\[
u(x,y) = \frac{x^4-6x^2y^2+y^4}{(x^2+y^2)^4}
\qquad
v(x,y)= \frac{4xy^3-4x^3y}{(x^2+y^2)^4}
\]
satisfy the Cauchy-Riemann equations for all $(x,y) \ne (0,0)$.
Show that $u, v$ are the real and imaginary parts of a holomorphic function $f : \C \setminus \{0\} \to \C$.
- Suppose that $f(z) = u(x,y)+iv(x,y)$ is holomorphic. Use the Cauchy-Riemann equations to show that both $u$ and $v$ satisfy Laplace's equation. That is, verify that
\[
\partial_1 (\partial_1 u) + \partial_2 (\partial_2 u) = 0
\qquad
\partial_1 (\partial_1 v) + \partial_2 (\partial_2 v) = 0
\]
both hold.
(The Laplacian of $g : \R^2 \to \R$ is $\triangle g = \partial_1 (\partial_1 g) + \partial_2 (\partial_2 g)$. Functions satisfying the Laplace equation $\triangle g = 0$ are called \define{harmonic}.)
- Let $f(z)= z^3$. Determine real-valued functions $u,v$ so that $f(x+iy) = u(x,y) + iv(x,y)$. Verify that both $u$ and $v$ satisfy the Laplace equation.
- Suppose $f(x+iy)=u(x,y)+iv(x,y)$ is holomorphic on $\C$. Suppose we know that $u(x,y) = x^5 - 10x^3y^2 + 5xy^4$. Use the Cauchy-Riemann equations to find all the possible forms of $v(x,y)$.
(The Cauchy Riemann equations have the following remarkable implication: suppose $f(z) = u(x,y)+iv(x,y)$ is holomorphic and we know a formula for $u$. Then we can recover $v$ up to a constant; similarly, if we know $v$ then we can recover $u$ up to a constant. Hence for holomorphic functions, the real part of a function determines the imaginary part up to constants, and vice versa.)
- Suppose that
\[
u(x,y) = x^3 -kxy^2 + 12xy - 12x
\]
for some constant $k \in \C$. Find all values of $k$ for which $u$ is the real part of a holomorphic function $f : \C \to \C$.
- Show that if $f : \C \to \C$ is holomorphic and $f$ has a constant real part then $f$ is constant.
- Show that the only holomorphic function $f : \C \to \C$ of the form $f(x+iy) = u(x) + iv(y)$ is given by $f(z) = \lambda z+a$ for some $\lambda \in \R$ and $a \in \C$.
- Suppose that $f(x+iy) = u(x,y) + iv(x,y)$. is a holomorphic function and that
\[
2 u(x,y) + v(x,y)= 5
\]
for all $x + iy \in \C$. Prove that $f$ is constant.
A Cauchy-Riemann Converse
- Let $f(z) = \sqrt{|xy|}$ where $z=x+iy$.
- Show from the definition that $f$ is not differentiable at the origin.
- Show however that the Cauchy-Riemann equations are satisfied at the origin. Why does this not contradict our converse to the Cauchy-Riemann equations?