\part{Advanced}
\chapter{Advanced theorems}
\section{Borel-Caratheodory theorem}
\section{Maximum modulus principle}
\section{Open mapping theorem}
\section{Rouche's theorem}

\chapter{Hardy spaces}
Applying ones knowledge of function analysis can be used to study holomorphic functions, specifically through \emph{Hardy spaces}.

\chapter{Multivariate complex analysis}


\chapter{Elliptic functions}
\section{Theta functions}


\chapter{Complex differential geometry}
\section{Möbius transformation}
\[f(z) = \frac{az+b}{cz+d}\]
\section{Schwarzian derivative}
\section{Schwarz lemma}