\part{Complex functions and differentiation}

\chapter{Complex functions}

We quickly consider complex function

- complex function
- decomposition of complex function
\begin{proposition}[Decomposition of complex functions]
For any complex function $f$, there exists 2 $\mathb{R}^2$ functions $u,v$ where the following holds.
\[ f(x+iy) = u(x,y) + iv(x,y) \]
\end{proposition}

Though we are assured that this function decomposition always exists, it is not always trivial to construct it. Regardless, this proposition is a fundamental trick for proofs in complex analysis since it links it to vector analysis by considering $\mathbb{R}^2$ functions.


\chapter{Complex diferentiable functions}

\section{Complex differentiable function}
\begin{definition}[Holomorphic function]
\end{definition}

\begin{definition}[Complex analytic function]
\end{definition}

\section{Cauchy-Riemann equations}
An extremely useful theorem in complex analysis that reformulates the definition of complex differentiability into a more interesting form.
\begin{theorem}[Cauchy-Riemann equations]
\end{theorem}

\section{Holomorphic function}

\begin{definition}[Holomorphic function]
\end{definition}

\begin{definition}[Entire function]
\end{definition}


