\section{Elliptic functions and related}

Particularly important in complex analysis.

\section{Weierstrass function}
\section{Dirichlet eta function}

\begin{definition}[Dirichlet eta function]
\[\eta(s) = \sum^{\infty}_{n=1} \frac{(-1)^{n-1}}{n^s}\]
\end{definition}
\section{Jacobi theta functions}

Jacobi theta functions unfortunately suffer from having incredibly inconsistent notation between different authors, so one must be cautious when consulting several texts. We will first define the \emph{basic Jacobi theta function}, which will be fundamental to developing other theta functions.

\begin{definition}[Basic Jacobi theta function]
\[\vartheta (z;\tau) = \sum^{\infty}_{n=-\infty} e^{i\pi n^2 \tau} e^{2 \pi i n z}\]
\end{definition}

Simple application of the Poisson summation formula gives the following result.

\begin{proposition}
Let $\tau > 0$, then the following holds
\[ \vartheta (0;\tau) = \tau^{-1/2}\vartheta (1 / \tau)\]
\end{proposition}


\begin{proposition}
\[ \pi^{-s /2} \Gamma (s/2) \zeta(s) = \frac{1}{2} \int^{\infty}_{0} t^{s/2 -1} (\vartheta(0;is)-1)ds\]
\end{proposition}