\part{Contour integration}

\chapter{Contour integral}
In vector analysis, there is the idea of a line integral; integrating along a differentiable curve in $\mathbb{R}^n$.

Complex analysis desires an analogous concept; the ability to integrate over a contour in $\mathbb{C}$.

- contour integral

\begin{definition}[Fundamental theorem of contour integration]
\end{definition}



\begin{theorem}[Fundamental theorem of contour integration]
\[\int_{\gamma} f'(z)dz = f(\gamma(1)) - f(\gamma(0))\]
\end{theorem}

\begin{proposition}[ML inequality]
\[ |\int_{\gamma} f(z)dz| \leq L(\gamma) \sup_{z\in \gamma} |f(z)| \]
\end{proposition}

\begin{proposition}[Jordan's lemma]
\end{proposition}

\chapter{Cauchy's integral theorem}

\section{Cauchy's integral theorem}
Perhaps the most important theorem within complex analysis is \emph{Cauchy's integral theorem}.

\begin{theorem}[Cauchy's integral theorem]
Let $f$ Let $\gamma$ be  
\[ \oint_{\gamma} f(z)dz = 0\]
\end{theorem}

\begin{theorem}[Cauchy's integral formula]
Let $f$ be holomorphic on $\Omega$ and $\gamma$ be a closed simple contour in $\Omega$, then we have the following.
\[f(z) = \frac{1}{2 \pi i} \oint_{\gamma} \frac{f(\xi)}{(\xi-z)} d\xi\]
Moreover, we have the following.
\[f^{(n)}(z) = \frac{n!}{2 \pi i} \oint_{\gamma} \frac{f(\xi)}{(\xi-z)^{n+1})} d\xi\]
\end{theorem}

Cauchy's integral formula paired with the ML inequality yields another useful inequality.

\begin{corollary}[Cauchy's estimate]
let $B(z,r)$ be an open ball in $\mathbb{C}$,
\[f^{(n)}(z) \leq \frac{n!}{r^n} \sup_{z\in \partial B(z,r)}|f(z)|\]
\end{corollary}

\begin{corollary}[Jordan's lemma]
\end{corollary}

\section{Consequences of Cauchy's integral theorem}


\begin{proposition}
Holomorphic functions are complex analytic.
\end{proposition}


\section{Liouville's formula}
- Liouville's theorem
\begin{theorem}[Liouville's theorem]
Let $f$ be an bounded, entire function, then $f$ is constant.
\end{theorem}

Liouville's theorem permits a particularly short and sweet proof for the fundamental theorem of algebra.

The fundamental theorem of algebra is actually a result more appropriate for complex analysis, however this name stuck since algegra was classically the study of polynomials. 

\begin{theorem}[Fundamental theorem of algebra (complex analysis)]
Let $P$ be a polynomial of degree $n$ with complex coefficients. Then there exists $c,r_i \in \mathbb{C}$ such that the following holds.
\[P(z) = c \prod^{n}_{i=1} (z-r_i)\]
\end{theorem}

- maximum modulus principle
- Analytic continuation
- Schwarz reflection principle
- Morera's theorem

