\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\# 6 (due February. 28)
}
\vskip.1in
  
%\endtitle
%\author
%\endauthor
%\endtopmatter

(1) Let $C$ be a closed, positive and simple curve. Using Green's theorem prove that $\frac{1}{2 i} \int_{C} \bar{z} d z=$ area enclosed by $C$

\vspace{.5cm}
(2) Consider the function $f(z)=(z+1)^{2}$ and the region $R$ bounded by the triangle with vertices $0,2, i$ (its boundary and interior). Find the points where $|f(z)|$ reaches its maximum and minimum value of $R$.
\vspace{.5cm}

(3) Find the maximum of $|\sin (z)|$ on $[0,2 \pi] \times[0,2 \pi]$.
\vspace{.5cm}
(4) Calculate:

(a) $\int_{0}^{2 \pi} \frac{d \theta}{a+b \cos (\theta)} \quad 0<b<a$.

HINT: Work backward using that $\cos (\theta)=\left(e^{i \theta}+e^{-i \theta}\right) / 2$ to convert the integral into a complex integral along the curve $|z|=1$

(b) $\int_{0}^{2 \pi} \frac{d \theta}{(a+b \cos (\theta))^{2}}$

(c) $\int_{0}^{2 \pi} \frac{\sin (\theta) d \theta}{(a+b \cos (\theta))^{2}}, \quad 0<b<a$.

\vspace{.5cm}

(5) Prove that if $f: \mathbb{C} \rightarrow \mathbb{C}$ is entire such that for some $n \in \mathbb{N}$
$$
\lim _{|z| \rightarrow \infty} \frac{|f(z)|}{|z|^{n}}=M<\infty
$$
then $f$ is a polynomial of degree at most $n$.

\vspace{.5cm}
(6) Let $A \subset \mathbb{C}$ be an open set and $f: A \rightarrow \mathbb{C}$ be an analytic function on $A$. Assuming that $z_{0} \in A$ such that
at $z_{0}$
$$
\left\{z \in \mathbb{C}:\left|z-z_{0}\right| \leq R\right\}, \quad R>0
$$
$f\left(z_{0}\right)=\frac{1}{\pi R^{2}} \iint_{\left|z-z_{0}\right| \leq R} f(x+i y) d x d y .$

\vspace{.5cm}

(7) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as
$$
f(x)=e^{-1 / x^{2}} \quad \text { if } \quad x \neq 0, \quad f(0)=0
$$
Show that $f$ is infinitely differentiable and $\forall n \in \mathbb{N} \quad f^{(n)}(0)=0$. Verify that the power serie of $f$ at $x=0$ does not agree with $f$ in any neighborhood of 0 .


\end{document}