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


Let $x$ be a complex variable of $\mathbb{C} \setminus \{\infty\}$.The function Sine Integral (noted $\operatorname{Si}$) is defined by the following third order differential equation
\begin{equation*}
\label{SI:diffeq}
\begin{split}
x \frac{\partial y (x)}{\partial x} + 2 \frac{\partial^{2} y (x)}{\partial x^{2}} + x \frac{\partial^{3} y (x)}{\partial x^{3}}& =0.
\end{split}\tag{SI.1.1}
\end{equation*}



Although $0$ is a singularity of SI.1.1, the initial conditions can be given by 
\begin{equation*}
\label{SI:inicond}
\begin{split}
\frac{\partial \frac{\operatorname{Si} (x)}{x}}{\partial x}& =1.
\end{split}\tag{SI.1.2}
\end{equation*}



Related function: \href{http://algo.inria.fr/esf/function/CI/CI.html#CI}{Cosine Integral}
\section*{SI.2 Series and asymptotic expansions}
\label{SI:asympt}
\subsection*{SI.2.1 Asymptotic expansion at $0$}
\label{743642323894624486}
\subsubsection*{SI.2.1.1 First terms}
\label{SI:asympt:0:termsec}



\begin{equation*}
\label{SI:asympt:0:terms}
\begin{split}
& \operatorname{Si} (x)\approx x \Biggl(1 - \frac{x^{2}}{18} + \frac{x^{4}}{600} - \frac{x^{6}}{35280} + \frac{x^{8}}{3265920} - \frac{x^{10}}{439084800} + \frac{x^{12}}{80951270400} -  \\
& \quad{}\quad{}\frac{x^{14}}{19615115520000}\ldots\Biggr).
\end{split}\tag{SI.2.1.1.1}
\end{equation*}

\subsubsection*{SI.2.1.2 General form}
\label{SI:asympt:0:genf}



\begin{equation*}
\label{SI:asympt:0:genfsum}
\begin{split}
& \operatorname{Si} (x)\approx x \sum_{n = 0}^{\infty} u (n) x^{n}.
\end{split}\tag{SI.2.1.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{SI:asympt:0:genfrec}
\begin{split}
u (n) (n + 1) \bigl(-1 - n + (n + 1)^{2}\bigr) + u (n - 2) (-1 + n)& =0.
\end{split}\tag{SI.2.1.2.2}
\end{equation*}
Initial conditions of SI.2.1.2.2 are given by
\begin{equation*}
\label{SI:asympt:0:genfic}
\begin{split}
u (1)& =0, \\
u (0)& =1, \\
u (2)& =-\frac{1}{18}.
\end{split}\tag{SI.2.1.2.3}
\end{equation*}
The recurrence SI.2.1.2.2 has the closed form solution
\begin{equation*}
\label{SI:asympt:0:RDINREFRDGENFROMRDCLOSED}
\begin{split}
u (2 n + 1)& =0, \\
u (2 n)& =\frac{(-1)^{n}}{(2 n + 1) \Gamma (2 n + 2)}.
\end{split}\tag{SI.2.1.2.4}
\end{equation*}

\subsection*{SI.2.2 Asymptotic expansion at $\infty$}
\label{743643747602464419}
\subsubsection*{SI.2.2.1 First terms}
\label{743643443707982742}



\begin{equation*}
\begin{split}
& \operatorname{Si} (x)\approx ser _{\Bigl[1,1,\Bigl[\Bigl[0,\Bigl[\Bigl[0,\frac{\pi}{2}\Bigr]\Bigr]\Bigr]\Bigr]\Bigr]} + \operatorname{e} ^{\biggl(-\frac{\operatorname{RootOf} _{\xi,2} (1 + \xi^{2})}{x}\biggr)} x y _{1} (x) +  \\
& \quad{}\quad{}\operatorname{e} ^{\biggl(-\frac{\operatorname{RootOf} _{\xi,1} (1 + \xi^{2})}{x}\biggr)} x y _{2} (x),
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
y _{0} (x)& =terms _{\Bigl[1,1,\Bigl[\Bigl[0,\Bigl[\Bigl[0,\frac{\pi}{2}\Bigr]\Bigr]\Bigr]\Bigr]\Bigr]} + \ldots \\
y _{1} (x)& =-\frac{1}{2} - \frac{\operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) x}{2} -  \\
& \quad{}\quad{}-\left(-\frac{1}{4} - \frac{5 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}}{4}\right) x^{2} - -\Biggl(-\operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) +  \\
& \quad{}\quad{}4 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \left(-\frac{1}{4} - \frac{5 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}}{4}\right)\Biggr) x^{3} + 2 \ldots \\
y _{2} (x)& =-\frac{1}{2} - \frac{\operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) x}{2} -  \\
& \quad{}\quad{}-\left(-\frac{1}{4} - \frac{5 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}}{4}\right) x^{2} - -\Biggl(-\operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) +  \\
& \quad{}\quad{}4 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \left(-\frac{1}{4} - \frac{5 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}}{4}\right)\Biggr) x^{3} + 2 \ldots
\end{split}
\end{equation*}

\subsubsection*{SI.2.2.2 General form}
\label{743643861773632661}
\paragraph*{SI.2.2.2.1 Auxiliary function $y _{0} (x)$}
\label{743643269843710796}
The auxiliary function $y _{0} (x)$ has the exact form 
\begin{equation*}
\begin{split}
y _{0} (x)& =\frac{\pi}{2}
\end{split}
\end{equation*}
\paragraph*{SI.2.2.2.2 Auxiliary function $y _{1} (x)$}
\label{743643248154420276}
The coefficients $u (n)$ of $y _{1} (x)$ satisfy the following recurrence
\begin{equation*}
\begin{split}
& 2 u (n) n + u (-1 + n) \Bigl(-2\operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) -  \\
& 5 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) (-1 + n) - 3 (-1 + n)^{2} \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr) +  \\
& u (n - 2) \bigl(8 - 5 n - 4 (n - 2)^{2} - (n - 2)^{3}\bigr)=0
\end{split}
\end{equation*}
whose initial conditions are given by
\begin{equation*}
\begin{split}
u (1)& =\frac{-\operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)}{2} \\
u (0)& =-\frac{1}{2}
\end{split}
\end{equation*}
\paragraph*{SI.2.2.2.3 Auxiliary function $y _{2} (x)$}
\label{743643351059198934}
The coefficients $u (n)$ of $y _{2} (x)$ satisfy the following recurrence
\begin{equation*}
\begin{split}
& 2 u (n) n + u (-1 + n) \Bigl(-2\operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) -  \\
& 5 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) (-1 + n) - 3 (-1 + n)^{2} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr) +  \\
& u (n - 2) \bigl(8 - 5 n - 4 (n - 2)^{2} - (n - 2)^{3}\bigr)=0
\end{split}
\end{equation*}
whose initial conditions are given by
\begin{equation*}
\begin{split}
u (0)& =-\frac{1}{2} \\
u (1)& =\frac{-\operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)}{2}
\end{split}
\end{equation*}
\end{document}