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\usepackage{amsmath, amsfonts, amssymb}
\usepackage{enumerate}

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\def\C{\mathbb{C}}
\def\D{\mathbb{D}}
\def\R{\mathbb{R}}
\def\N{\mathbb{N}}

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\def\la{\langle}
\def\ra{\rangle}
\def\half{\frac{1}{2}}
\def\LRA{\Leftrightarrow}

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\renewcommand{\Re}{\text{Re}}
\renewcommand{\Im}{\text{Im}}


\title{Math 185: Homework 1 Solution}
\author{Instructor: Peng Zhou}
\date{August 2020}

\begin{document}

\maketitle
The following exercises are from Stein's textbook, Chapter 1. 

\section{Exercise 2}
Let $\la \cdot, \cdot \ra$ denote the usual product in $\R^2$. In other words, if $Z=(x_1,y_1)$ and $W=(x_2, y_2)$, then 
$$ \la Z, W \ra = x_1 x_2 + y_1 y_2 $$
Similarly, we may define a Hermitian inner product $( \cdot, \cdot)$ on $\C$ by 
$$ (z, w) = z \bar w. $$
The term Hermitian is used to describe the fact that $( \cdot, \cdot )$ is not symmetric but rather satisfies the relation
$$ (z,w) = \overline{(w,z)}. $$
Show that
$$ \la z,w \ra = \half [(z,w) + (w,z)] = \Re ( z, w ) $$
where we used the usual identification $z = x+iy \in \C$ with $(x,y) \in \R^2$. 
 

\section{Exercise 7}
The family of mappings introduced here plays an important role in complex analysis. These mappings, sometimes called Blaschke factors, will reappear in various applications in later chapters. 
\begin{enumerate}[(a)]
    \item Let $z,w$ be two complex numbers such that $\bar z w \neq 1$. Prove that 
    $$ \left| \frac{w -z}{1- \bar w z} \right| < 1 \quad \text{if  $|z|<1$ and  $|w| < 1$} $$
    and also that 
    $$ \left| \frac{w -z}{1- \bar w z} \right| = 1 \quad \text{if  $|z|=1$ or  $|w| = 1$} $$
    \item Prove that for a fixed $w$ in the unit disk $\D$, the mapping
    $$ F: z \mapsto \frac{w-z}{1- \bar w z} $$
    satisfies the following conditions: 
    \begin{enumerate}[(i)]
        \item $F$ maps the unit disk to itself, and is holomorphic.
        \item $F$ interchanges $0$ and $w$, namely $F(0) = w$ and $F(w)=0$.
        \item $|F(z)|=1$ if $|z|=1$. 
        \item $F: \D \to \D$ is bijective. 
    \end{enumerate} 
\end{enumerate}
 \section{Exercise 16 (a) (c) (e)}
Determine the radius of convergence of the series $\sum_{n=1}^\infty a_n z^n$ when 
\begin{enumerate}
    \item[(a)] $a_n = (\log n)^2$
    \item[(c)] $a_n = \frac{n^2}{4^n + 3n}$
    \item [(e)] Find the radius of convergence for the hypergeometric series
    $$F(\alpha, \beta, \gamma; z) = 1 + \sum_{n=1}^\infty \frac{\alpha (\alpha+1) \cdots (\alpha + n-1) \beta (\beta+1) \cdots (\beta+n-1)}{n!  \gamma (\gamma+1) \cdots (\gamma+n-1)} z^n $$
    Here $\alpha, \beta \in \C$ and $\gamma \neq 0, -1, -2, \cdots $. 
\end{enumerate}
 
\section{Exercise 17}
Show that if $\{a_n\}_{n=0}^\infty$ is a sequence of non-zero complex numbers such that 
$$ \lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|} = L $$
then 
$$ \lim_{n \to \infty} |a_n|^{1/n} = L.  $$
 
\section{Exercise 22}
Let $\N = \{1, 2, \cdots, \}$ denote the set of positive integers. A subset $S \subset \N$ is said to be in arithematic progression if 
$$ S = \{a, a+d, a+2d, \cdots \} $$
for some $a, d \in \N$. Here $d$ is called the step of $S$. Show that $\N$ cannot be partitioned into finite number of arithematic progressions with distinct step sizes. 
 

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