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\date{April 2012}
\title{M.S and M.A Comprehensive Analysis Exam}
\author{Alessandro Arsie and Henry Wente}
\begin{document}
\maketitle

\textbf{To get full credit you must show all your work.}

This exam contains 6 real analysis and 6 complex variables questions.

\section*{Real Analysis}
$100\%$ will be obtained for complete answers to {\bf four} questions out of six. Indicate
clearly which four questions you wish to be graded. 
\begin{enumerate}
 \item Let $\{f_n\}$ be a sequence of continuous real-valued functions on the closed interval $[a,b]$. 
\begin{enumerate}
        \item Define what it means for $f_n$ to converge uniformly to a function $f$ on $[a,b]$ and prove that if $f_n$ converges uniformly to $f$, then the limit function $f$ is continuous. 
        \item Give an example of a sequence of continuous functions $\{f_n\}$ on $[0,1]$ which converges pointwise to a function $f$ on $[0,1]$ and which is not continuous there. 
       \end{enumerate}

 \item Let $f$ be a continuous real-valued function on $[a,b]$ and suppose that $$\int_a^b x^n f(x)\; dx=0, \quad \forall n\geq 0.$$
Show that $f(x)$ is identically zero. 

\item Let $E\subset X$ be a compact subset in the metric space $(X,d)$. For $p\in X$ define 
$$d(p,E):=\inf\{d(p,q)| q\in E\}.$$
Show that for each fixed $p$, there exists a $q^*\in E$ with $d(p,E)=d(p,q^*)$. 
\item Let $f$ be a continuous real-valued function on $[a,b]$. Suppose $f(x)\geq 0$ and that there is an $x_0\in [a,b]$ with $f(x_0)>0.$ 
Show that 
$$\int_a^b f(x)\; dx>0.$$

\item Let $f$ be a continuous real-valued function on $[a,b]$. Show that $f$ is uniformly continuous there.  

         
\item Suppose the sequence $\{f_n\}$ is converging uniformly to $f$ on $[a,b]$, where $f_n$ and $f$ are continuous.
\begin{itemize}
\item Prove that
$$\int_a^b f(x)\; dx=\lim_{n\rightarrow +\infty}\int_a^b f_n(x)\;dx.$$
What about 
$$\lim_{n\rightarrow+\infty}\int_a^b \sin(f_n(x))\; dx\;\; ?$$
\item  Give an example of continuous functions $\{f_n\}$ and $f$ where $f_n$ converges pointwise to $f$ on $[0,1]$ and yet
$$\int_0^1 f(x)\; dx\neq \lim_{n\rightarrow +\infty} \int_0^1 f_n(x)\; dx.$$
\end{itemize}

\end{enumerate}


\section*{Complex Analysis}
$100\%$ will be obtained for complete answers to {\bf four} questions out of six. Indicate
clearly which four questions you wish to be graded. 
\begin{enumerate}

\item Given the function $$f(z)=\frac{z}{(z-2)(z+i)}$$
determine where it is analytic and where its singularities are located. Determine also the type of singularities. 
Finally expand it in a Laurent series in the following regions: 
\begin{itemize}
 \item $|z|<1$;
\item $1<|z|<2.$
\end{itemize}



\item State and prove Cauchy's integral formula. 

\item  Evaluate the following integral, where $C$ is a simple closed curve going around the origin counterclockwise:
$$\oint_C \frac{e^{z^2}}{3z^2}\; dz$$

\item The complex logarithmic function:
\begin{itemize}
 \item  Define $\log(z)$;
\item  Use the definition and Cauchy-Riemann equations to prove that $\log(z)$ is an analytic function where it is defined. 
\item Show that is multi-valued, and compute all determinations of $\log(2i)$. 
\end{itemize}


\item Let $C$ be an open upper semicircle of radius $R$ with its center at the origin, and suppose $R>a>0$. 
\begin{itemize}
 \item Let $f(z)=\frac{1}{z^2+a^2}$. Show that {\bf on $C$} 
$$|f(z)|\leq \frac{1}{R^2-a^2}$$ and that 
$$\left|\int_C f(z)\; dz\right|\leq \frac{\pi R}{R^2-a^2}.$$
\item Use the previous result and Residue Theorem to compute 
$$ \int_{-\infty}^{\infty}\frac{1}{x^2+a^2}.$$
\end{itemize}


\item Find the location of branch points and discuss branch cuts for the following function: 
$$f(z)=\left((z-1)(z+1)\right)^{\frac{1}{2}}.$$

\end{enumerate}


\end{document}