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\begin{document}
\chapter*{HSN Hyperbolic Sine}
\label{HSN}
\section*{HSN.1 Introduction}
\label{HSN:intro}


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



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



Related function: \href{http://algo.inria.fr/esf/function/HCS/HCS.html#HCS}{Hyperbolic Cosine}
\section*{HSN.2 Series and asymptotic expansions}
\label{HSN:asympt}
\subsection*{HSN.2.1 Taylor expansion at $0$}
\label{743622933674766980}
\subsubsection*{HSN.2.1.1 First terms}
\label{743622663646261121}



\begin{equation*}
\label{HSN:asympt:0:RDLBLRDTERMSRDEQ}
\begin{split}
\operatorname{sinh} (x)& =x + \frac{1}{6}   x^{3} + \frac{1}{120}   x^{5} + \frac{1}{5040}   x^{7} + \frac{1}{362880}   x^{9} + \frac{1}{39916800}   x^{11} +  \\
& \quad{}\quad{}\frac{1}{6227020800}   x^{13} + \frac{1}{1307674368000}   x^{15} + \operatorname{O} \bigl(x^{16}\bigr).
\end{split}\tag{HSN.2.1.1.1}
\end{equation*}

\subsubsection*{HSN.2.1.2 General form}
\label{743622380977721667}



\begin{equation*}
\label{HSN:asympt:0:RDLBLRDGENFORMRDEQ}
\begin{split}
\operatorname{sinh} (x)& =\sum_{n = 0}^{\infty} u (n) x^{n}.
\end{split}\tag{HSN.2.1.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{HSN:asympt:0:toto}
\begin{split}
-u (n) + \bigl(n^{2} + 3 n + 2\bigr) u (n + 2)& =0.
\end{split}\tag{HSN.2.1.2.2}
\end{equation*}
Initial conditions of HSN.2.1.2.2 are given by
\begin{equation*}
\label{HSN:asympt:0:RDLBLRDGENFORMRDIC}
\begin{split}
u (0)& =0, \\
u (1)& =1.
\end{split}\tag{HSN.2.1.2.3}
\end{equation*}
The recurrence HSN.2.1.2.2 has the closed form solution
\begin{equation*}
\label{HSN:asympt:0:RDLBLRDGENFORMRDCLOSED}
\begin{split}
u (2 n + 1)& =\frac{1}{\Gamma (2 n + 2)}, \\
u (2 n)& =0.
\end{split}\tag{HSN.2.1.2.4}
\end{equation*}

\subsection*{HSN.2.2 Asymptotic expansion at $\infty$}
\label{743622951446693476}
\subsubsection*{HSN.2.2.1 Exact form}
\label{743622261166304676}



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

\section*{HSN.3 Graphs}
\label{743630601711169923}
\subsection*{HSN.3.1 Real axis}
\label{743630196898955214}
\begin{center}
\includegraphics[width=6cm]{HSN/744692266725078583}
\end{center}
\subsection*{HSN.3.2 Complex plane}
\label{743630888262887414}
\begin{center}
\includegraphics[width=6cm]{HSN/744692803619385287}
\end{center}
\end{document}