\part{Advanced}

\chapter{Advanced theorems}

There are many assorted results that give deeper insight to the nature of holomorphic functions, building from the basic results in the fundamental part of this book.

\section{Rouche's theorem}

\section{Open mapping theorem}

Recall the definition of an open map from general topology.
Holomorphic functions are topologically special not only because they are continuous functions, but also open maps! THat is, open subsets of the domain are mapped to open subsets of the image (and vice versa due to continuity).

\begin{theorem}[Open map theorem]
Let $f$ be nonconstant and holomorphic, then $f$ is an open map
\end{theorem}

\subsection{Maximum modulus principle}

The nature of open sets of $\mathbb{C}$ and its subspaces


\begin{theorem}[Maximum modulus principle]
Let $f$ be a nonconstant and holomorphic, then the image of $|f|$ has no greatest element.
\end{theorem}





\section{Picard's theorems}

\begin{theorem}[Picard's little theorem]
	Let $f$ be entire, then the image of $f$ is $\mathbb{C}$ or $\mathbb{C}\setminus \{z_0\}$
\end{theorem}
\begin{theorem}[Picard's great theorem]
\end{theorem}












\section{Factorization theorems}

\subsection{Elementary factors}
\subsection{Weierstrass factorization theorem}
\subsection{Hadamard factorization theorem}
Hadamard extends on 

\section{Proof of Cauchy's theorem}

Though we have been using Cauchy's theorem heavily in this book already, we are yet to offer a proof of its ultimate version; this requires some topological language to discuss, specifically that of homotopies (discussed in algebraic topology).

\section{Borel-Caratheodory theorem}

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





\chapter{Conformal mappings}

Area of complex analysis with high geometric significance.

\section{Möbius transformation}
\[f(z) = \frac{az+b}{cz+d}\]
\section{Riemann mapping theorem}
\section{Montel'stheorem}
\section{Schwarzian derivative}
\section{Schwarz lemma}