\documentclass[12pt]{amsart}
\title[]{Mid Term 3}
\author{}

\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb}
\usepackage[all]{xy}
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\usepackage{graphicx}
\newcommand{\ds}{\displaystyle}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\cosec}{\operatorname{cosec}}
\renewcommand{\iota}{{\mathbf i}}
\newcommand{\ol}[1]{\overline{#1}}
\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\Res}[2]{\underset{#1}{\operatorname{Res}}\left(#2\right)}
\newcommand{\PVint}{\operatorname{P.V.}\int}

\newcommand{\Ann}{\operatorname{Ann}}
\newcommand{\problem}[2]{\noindent {\bf Problem #1. (#2 points) }}
\newcommand{\PP}[2]{\noindent {\bf (#1) #2.--}}
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\newtheorem*{defn}{Definition}
\newtheorem*{rem}{Remark}
\newtheorem*{rems}{Remarks}
\newtheorem*{ex}{Example}

\begin{document}
\maketitle

\problem{1}{10} What is the radius of convergence of
$\ds\sum_{n=1}^{\infty} \frac{n!}{n^n} z^n$.\\

\problem{2}{20} Consider the function $\ds f(z) = \frac{z}{\sin(z)}$.\\
\begin{enumerate}
\item Prove that $z=0$ is a removable singularity of $f(z)$.\\
\begin{center}
{\em Now, let $\ds\frac{z}{\sin(z)} = \sum_{n=0}^{\infty} c_n z^n$ be the
Taylor series expansion of $f(z)$ centered at $0$.}\\
\end{center}

\item What is the radius of convergence of $\ds\sum_{n=0}^{\infty}
c_nz^n$?\\

\item Prove that $c_{2k+1}=0$ for every $k\geq 0$.\\

\item Compute $c_0,c_2$ and $c_4$.\\
\end{enumerate}

\problem{3}{10} Compute $\ds\Res{z=0}{\frac{e^{3z}-3e^z+2}{z^4}}$.\\

\problem{4}{10} Let $C$ be the counterclockwise circle
of radius $3$, centered at $0$. Prove that
$\ds \int_C \frac{2z^3+1}{z^2(z-1)(z-2)}\, dz = 4\pi\iota$.\\

\problem{5}{10} Compute the Laurent series expansion of
$\ds \frac{1}{z^2(z-2)}$ near $z=2$.\\

\problem{6}{20} Prove that
$\ds\int_0^{2\pi} \frac{d\theta}{(2+\cos(\theta))^2}
= \frac{4\pi}{3\sqrt{3}}$.\\

\problem{7}{20} Prove that
$\ds \int_0^{\infty} \frac{x\sin(x)}{x^2+1}\, dx = \frac{\pi}{2e}$.\\

\noindent {\bf Bonus.} Let $a\in\R_{>0}$ be a fixed positive
real number. Prove that $\ds \int_0^\pi \tan(x+a \iota)\, dx
= \pi\iota$.\\

%\noindent {\bf Bonus.} Let $m\in\Z_{\geq 0}$ be a non--negative
%integer. Prove that
%$\ds \int_0^\pi \cos^{2m}(\theta)\, d\theta = 
%\frac{\pi}{2^{2m}}\frac{(2m)!}{(m!)^2}$.

\end{document}
