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% \title{New Graduate Analysis A \& B}
% \author{Dana P. Williams}
% \date{1 May 2010}
% \maketitle
%\newpage
\begin{center}
\textsc{Math 103 Syllabus (cross-listed with Math 73)}
\end{center}
\begin{enumerate}
\item Abstract Measure Theory (12 Lectures)
\begin{enumerate}
\item Measures, $\sigma$-algebras and all that.
\item An Example: Lebesgue measure on $\mathbf{R}$ and/or
$\mathbf{R}^{n}$.
% \item Getting a measure from an outer measure \& the Carath\'eodory
% Theorem.
\item Integration in an abstract measure space
\item The convergence Theorems and applications.
\item Product measures, Tonelli and Fubini.
\item Sources
\begin{enumerate}
\item We have in mind cherry picking from
Rudin's \emph{Real \& Complex}
(Chapters 2, 6 and 8)
since
that will be the usual reference for the second part of the
course. Instructors will have to develop Lebesgue measure on
their own or possibly using Royden \& Fitzpatrik as a guide.
\item Obviously, time constraints and the instructor's interests
will dictate what topics can be covered and at what depth. The
topologists would love some attention paid to Lebesgue measure
in $\mathbf{R}^{n}$ at some point.
\end{enumerate}
\end{enumerate}
\item Complex Analysis (15 Lectures)
\begin{enumerate}
\item Elementary Properties
\begin{enumerate}
\item Complex differentiation, Cauchy Riemann equations and path integrals
\item Local Cauchy Theorem
\item Holomorphic implies analytic
\item Global Cauchy Theorem
\item Sources
\begin{enumerate}
\item The basic source we have in mind is Chapter 10 of Rudin's
\emph{Real \& Complex}. This can be followed
fairly closely --- even if it is fairly sophisticated.
\item Dana was taught that in an outline, there had to always be at
least two sub-parts under any given item.
\end{enumerate}
\end{enumerate}
\item Selected Topics --- Lecturer's Discretion
\begin{enumerate}
\item Maximum Modulus
\item Isolated Singularities and Laurent Series
\item Residue Theorem and Applications
\item Argument Principle and Roche's Theorem
\item Normal Families and Riemann Mapping Theorem
\item Sources
\begin{enumerate}
\item When Dana tried this before, he picked and chose from
Chapter's 12--14 of Rudin's
\emph{Real \& Complex}.
\item Sample Goal: try to build up enough background to at least
pretend to prove Theorem~13.11 (Rudin) (at least
$(b)\Longleftrightarrow (c) \Longleftrightarrow
(d)\Longleftrightarrow (f)$). That can't be done without
leaving the proofs of some of the harder results.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{document}
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