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\begin{document}
\chapter*{HCS Hyperbolic Cosine}
\label{HCS}
\section*{HCS.1 Introduction}
\label{HCS:intro}


Let $x$ be a complex variable of $\mathbb{C} \setminus \{\infty\}$.The function Hyperbolic Cosine (noted $\operatorname{cosh}$) is defined by the following second order differential equation
\begin{equation*}
\label{HCS:diffeq}
\begin{split}
-y (x) + \frac{\partial^{2} y (x)}{\partial x^{2}}& =0.
\end{split}\tag{HCS.1.1}
\end{equation*}



The initial conditions of HCS.1.1 are given at $0$ by
\begin{equation*}
\label{HCS:inicond}
\begin{split}
\operatorname{cosh} (0)& =1, \\
\frac{\partial \operatorname{cosh} (x)}{\partial x} (0)& =0.
\end{split}\tag{HCS.1.2}
\end{equation*}



Related function: \href{http://algo.inria.fr/esf/function/HSN/HSN.html#HSN}{Hyperbolic Sine}
\section*{HCS.2 Series and asymptotic expansions}
\label{HCS:asympt}
\subsection*{HCS.2.1 Asymptotic expansion at $\infty$}
\label{743613193291319484}
\subsubsection*{HCS.2.1.1 Exact form}
\label{743613841618109973}



\begin{equation*}
\begin{split}
\operatorname{cosh} (x)& =\frac{\operatorname{e} ^{x}}{2} - \frac{\operatorname{e} ^{(-x)}}{2}.
\end{split}
\end{equation*}

\subsection*{HCS.2.2 Taylor expansion at $0$}
\label{743613440951262229}
\subsubsection*{HCS.2.2.1 First terms}
\label{743613331720217518}



\begin{equation*}
\label{HCS:asympt:0:RDLBLRDTERMSRDEQ}
\begin{split}
\operatorname{cosh} (x)& =1 + \frac{1}{2}   x^{2} + \frac{1}{24}   x^{4} + \frac{1}{720}   x^{6} + \frac{1}{40320}   x^{8} + \frac{1}{3628800}   x^{10} +  \\
& \quad{}\quad{}\frac{1}{479001600}   x^{12} + \frac{1}{87178291200}   x^{14} + \operatorname{O} \bigl(x^{16}\bigr).
\end{split}\tag{HCS.2.2.1.1}
\end{equation*}

\subsubsection*{HCS.2.2.2 General form}
\label{743613112620824941}



\begin{equation*}
\label{HCS:asympt:0:RDLBLRDGENFORMRDEQ}
\begin{split}
\operatorname{cosh} (x)& =\sum_{n = 0}^{\infty} u (n) x^{n}.
\end{split}\tag{HCS.2.2.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{HCS:asympt:0:toto}
\begin{split}
-u (n) + \bigl(n^{2} + 3 n + 2\bigr) u (n + 2)& =0.
\end{split}\tag{HCS.2.2.2.2}
\end{equation*}
Initial conditions of HCS.2.2.2.2 are given by
\begin{equation*}
\label{HCS:asympt:0:RDLBLRDGENFORMRDIC}
\begin{split}
u (1)& =0, \\
u (0)& =1.
\end{split}\tag{HCS.2.2.2.3}
\end{equation*}
The recurrence HCS.2.2.2.2 has the closed form solution
\begin{equation*}
\label{HCS:asympt:0:RDLBLRDGENFORMRDCLOSED}
\begin{split}
u (2 n)& =\frac{1}{\Gamma (2 n + 1)}, \\
u (2 n + 1)& =0.
\end{split}\tag{HCS.2.2.2.4}
\end{equation*}

\section*{HCS.3 Graphs}
\label{743621966378204911}
\subsection*{HCS.3.1 Real axis}
\label{743621989909304114}
\begin{center}
\includegraphics[width=6cm]{HCS/744672745895430739}
\end{center}
\subsection*{HCS.3.2 Complex plane}
\label{743621263045678313}
\begin{center}
\includegraphics[width=6cm]{HCS/744672249440828}
\end{center}
\end{document}