\part{Complex functions and differentiation}

\chapter{Complex functions}

Much of this chapter will be porting many ideas from real analysis to complex analysis. We quickly consider complex function.

\begin{definition}[Complex function]
\[ f : \mathbb{C} \to \mathbb{C} \]
\end{definition}

\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.

\section{}

\section{Complex differentiable function}


\begin{definition}[Derivative]
Let $f$ be a complex function, the \emph{complex derivative of $f$ at $z_0$} is the following limit.
\[ f'(z_0) = \lim_{z \to z_0} \frac{f(z)-f(z_0)}{z-z_0}\]
\end{definition}



\begin{definition}[Complex differentiable function]
A complex function is \emph{complex differentiable at $z_0$} iff the derivative of $f$ at $z_0$ converges to a finite value.
\[ f'(z_0) = \lim_{z \to z_0} \frac{f(z)-f(z_0)}{z-z_0}\]
A complex function is said to be complex differentiable on $U$ iff it is complex differentiable for every $u \in U$.
\end{definition}

On an initial encounter, this is no different to the definition of a real differentiable function. Indeed the only difference is the fact that it works on complex functions rather than real functions, but there is a subtle side effect; limits in $\mathbb{C}$ differ from those in $\mathbb{R}$ since unlike in $\mathbb{R}$, one can approach a point from various directions along mixtures of the real and imaginary axis.


When a function is complex differentiable, we can leverage many familiar rules from real analysis in calculating the complex derivative
\begin{itemize}
\item Linearity of the derivative
\item Product rule
\item Quotient rule
\item Chain rule
\item Inverse function theorem
\end{itemize}


\section{Cauchy-Riemann equations}
An extremely useful theorem in complex analysis that reformulates the definition of complex differentiability as a statement on its decomposition is the following set of equations.

\begin{theorem}[Cauchy-Riemann equations]
Let $f=u+iv$, $f$ is complex differentiable at $x+iy$ iff $u,v$ are continuously differentiable and the following PDEs hold.
	\[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \]
	\[ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \]
	\[\]
\end{theorem}

The reason the Cauchy-Riemann equations are so powerful is because they link the total differentiability of  $u,v$ to the complex differentiability of $f$, allowing for the methods of vector analysis to be used in the realm of complex analysis (the inverse function theorem depends on this).

\section{Holomorphic function}

Often it is interesting when we have complex differentiability over an open set; this is the idea of holomorphic functions. We will be able to construct many important theorems with this notion later on.

\begin{definition}[Holomorphic function]
A complex function is \emph{holomorphic on $\Omega$} iff it is complex differentiable at every point within the open set $\Omega$.
A complex function holomorphic on its entire domain is a \emph{holomorphic function}
\end{definition}

\begin{definition}[Entire function]
An \emph{entire function} is a holomorphic function on $\mathbb{C}$.
\end{definition}


\begin{definition}[Complex analytic function]
A complex function is \emph{complex analytic on $\Omega$} iff for any $z_0 \in \Omega$, the Taylor series of $f$ converges to $f$ in a neighborhood of $z_0$.
A complex function that is complex analytic on its entire domain is a \emph{complex analytic function}.
\end{definition}