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\centerline{{\it  Class 32} {\bf (Monday April 13)}}
 
\begin{description}
\item[SUMMARY] Review for Second Exam
\end{description}

Here is a brief summary of all the topics that we have considered in this second half of the course:
\begin{itemize}
\item complex functions of a real variable
\item Arcs, curves, contours
\item Parametrization 
\item Differentiation and antidifferentiation of complex functions of a real variable
\item Contour integration
\item $\displaystyle{\oint (z-z_0)^n dz = 2 \pi i \delta_{n,-1}}$
\item Antiderivatives of analytic functions
\item Cauchy-Goursat Theorem
\item Deformation Invariance Theorem
\item Path Independence Theorem
\item Simply and Multiply Connected Domains
\item Cauchy Integral Formula
\item Generalized Cauchy Integral Formula
\item Liouville's Theorem
\item Cauchy's Inequality
\item Maximum and minimum modulus theorem(s)
\item Analytic functions are infinitely differentiable
\item Zeroes and poles
\item Classification of singularities (isolated, removable, branch and essential)
\item Residues
\item Cauchy's Residue Theorem
\item Laurent Series
\item Applications of Laurent Series to Contour Integration
\item Cauchy's Second Residue Theorem
\item Applications of Residues to Real Trigonometric Integrals
\item Applications of Residues to Real Improper Integrals (Jordan's Lemma)
\end{itemize}

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