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


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



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



Related function: \href{http://algo.inria.fr/esf/function/CS/CS.html#CS}{Cosine}
\section*{SN.2 Series and asymptotic expansions}
\label{SN:asympt}
\subsection*{SN.2.1 Asymptotic expansion at $\infty$}
\label{743644295397149192}
\subsubsection*{SN.2.1.1 Exact form}
\label{74364416665122933}



\begin{equation*}
\begin{split}
\operatorname{sin} (x)& =-\frac{i}{2}\operatorname{e} ^{\bigl(-\operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) x\bigr)} + \frac{i}{2} \operatorname{e} ^{\bigl(-\operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) x\bigr)}.
\end{split}
\end{equation*}

\subsection*{SN.2.2 Taylor expansion at $0$}
\label{743644308731605735}
\subsubsection*{SN.2.2.1 First terms}
\label{743644678125671752}



\begin{equation*}
\label{SN:asympt:0:RDLBLRDTERMSRDEQ}
\begin{split}
\operatorname{sin} (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{SN.2.2.1.1}
\end{equation*}

\subsubsection*{SN.2.2.2 General form}
\label{743644420981171466}



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

\section*{SN.3 Graphs}
\label{743654294830536553}
\subsection*{SN.3.1 Real axis}
\label{743654504270601868}
\begin{center}
\includegraphics[width=6cm]{SN/744805122936725520}
\end{center}
\subsection*{SN.3.2 Complex plane}
\label{74365468147354888}
\begin{center}
\includegraphics[width=6cm]{SN/744805238128967060}
\end{center}
\end{document}