\documentclass[12pt]{article}
\usepackage{framed}
\usepackage{hyperref}
\usepackage{amssymb}
\usepackage{amsmath,amssymb}
\usepackage{color}
\setlength{\topmargin}{-8mm}
\setlength{\textheight}{55pc}
\setlength{\textwidth}{16cm}
%\setlength{\oddsidemargin}{-5mm}
%\setlength{\evensidemargin}{-5mm}
\begin{document}
\pagestyle{empty}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% macros for this file                       %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\topmatter
%\title
{\bf Math 122-A \hskip.1in   Winter 2022\hskip.1in HOMEWORK\# 5 (due February. 17)
}
\vskip.1in
  
%\endtitle
%\author
%\endauthor
%\endtopmatter

\begin{enumerate}
    \item  Let $z_{0} \in \mathbb{C}$ be any interior point to any positive oriented simple closed curve $C$. Prove
$$
\oint_{C} \frac{d z}{z-z_{0}}=2 \pi i, \quad \oint_{C} \frac{d z}{\left(z-z_{0}\right)^{n+1}}=0, n=1,2,3, \ldots
$$


\item  Let $C$ be the contour of the circle $|z-i|=2$ in the positive sense. Find

\begin{enumerate}


\item  ${\displaystyle\oint_{C} \frac{d z}{z^{2}+4},}$
\item  ${\displaystyle \oint_{C} \frac{e^{z} d z}{z-\pi i / 2}}$,
\item ${\displaystyle \oint_{C} \dfrac{\cos (z) d z}{\left(z^{2}+16\right) z}}$
\item  ${\displaystyle\oint_{C} \dfrac{d z}{2 z+1}}$.
\end{enumerate}
\item For $z \in \mathbb{C}$ with $|z| \neq 3$, denote $C$ the contour of the circle $|z|=3$ in the positive sense and define
$$
g(z)=\oint_{C} \frac{2 w^{2}-w-2}{w-z} d w .
$$
Find the values of $g(2)$ and $g(3+2 i)$.
\item  Assuming that the given contour ids positive oriented, compute

\begin{enumerate}
    

\item  ${\displaystyle\oint_{|z|=3} \frac{\left(e^{z}+z\right) d z}{z-2}}$,
\item ${\displaystyle\oint_{|z|=1} \frac{e^{z} d z}{z^{2}}}$, 
\item ${\displaystyle\oint_{|z|=2} \frac{d z}{z^{2}+z+1}}$,
\item ${\displaystyle\oint_{|z|=1} \frac{d z}{z^{2}-1}}$.

\end{enumerate}
DEFINITION: A $f: \mathbb{C} \rightarrow \mathbb{C}$ is an ENTIRE function if $f$ is analytic in all $\mathbb{C}$.

\item Prove that if $f$ is entire and there exist $z_{0} \in \mathbb{C}$ and $r>0$ such that
$$
f(\mathbb{C}) \cap\left\{z \in \mathbb{C}:\left|z-z_{0}\right|<r\right\}=\emptyset
$$
then $f$ is a constant function.
\item Identify all entire functions $f$ such that $\forall z \in \mathbb{C} \quad|f(z)| \leq 2|z|$.
    
    
    
    
    
    
    %%%%%%%%%%
\end{enumerate}




\end{document}