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


Let $x$ be a complex variable of $\mathbb{C} \setminus \{\infty\}$.The function Hyperbolic Sine Integral (noted $\operatorname{Shi}$) is defined by the following third order differential equation
\begin{equation*}
\label{SHI: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{SHI.1.1}
\end{equation*}



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



Related function: \href{http://algo.inria.fr/esf/function/CHI/CHI.html#CHI}{Hyperbolic Cosine Integral}
\section*{SHI.2 Series and asymptotic expansions}
\label{SHI:asympt}
\subsection*{SHI.2.1 Asymptotic expansion at $\infty$}
\label{743641582086890698}
\subsubsection*{SHI.2.1.1 First terms}
\label{743640755308773384}



\begin{equation*}
\begin{split}
& \operatorname{Shi} (x)\approx  \\
& \quad{}\quad{}ser _{\Bigl[1,1,\Bigl[\Bigl[0,\Bigl[\Bigl[0,\frac{\pi}{2}\Bigr]\Bigr]\Bigr]\Bigr]\Bigr]} + \operatorname{e} ^{\frac{1}{x}} x y _{1} (x) + \operatorname{e} ^{\Bigl(-\frac{1}{x}\Bigr)} 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{x}{2} + x^{2} + 3 x^{3} + 2 \ldots \\
y _{2} (x)& =\frac{1}{2} - \frac{x}{2} + x^{2} - 3 x^{3} + 2 \ldots
\end{split}
\end{equation*}

\subsubsection*{SHI.2.1.2 General form}
\label{743641107998164140}
\paragraph*{SHI.2.1.2.1 Auxiliary function $y _{0} (x)$}
\label{743640316472625851}
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*{SHI.2.1.2.2 Auxiliary function $y _{1} (x)$}
\label{743640253130639625}
The coefficients $u (n)$ of $y _{1} (x)$ satisfy the following recurrence
\begin{equation*}
\begin{split}
& -2u (n) n + u (n - 1) \bigl(-3 + 3 (n - 1)^{2} + 5 n\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{1}{2} \\
u (0)& =\frac{1}{2}
\end{split}
\end{equation*}
This recurrence has the closed form solution
\begin{equation*}
\begin{split}
u (n)& =\frac{\Gamma (n + 1)}{2}.
\end{split}
\end{equation*}
\paragraph*{SHI.2.1.2.3 Auxiliary function $y _{2} (x)$}
\label{743641508552854512}
The coefficients $u (n)$ of $y _{2} (x)$ satisfy the following recurrence
\begin{equation*}
\begin{split}
& -2u (n) n + u (n - 1) \bigl(3 - 3 (n - 1)^{2} - 5 n\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{1}{2}
\end{split}
\end{equation*}
This recurrence has the closed form solution
\begin{equation*}
\begin{split}
u (n)& =\frac{(-1)^{n} \Gamma (n + 1)}{2}.
\end{split}
\end{equation*}
\subsection*{SHI.2.2 Asymptotic expansion at $0$}
\label{743639718148395497}
\subsubsection*{SHI.2.2.1 First terms}
\label{SHI:asympt:0:termsec}



\begin{equation*}
\label{SHI:asympt:0:terms}
\begin{split}
& \operatorname{Shi} (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{SHI.2.2.1.1}
\end{equation*}

\subsubsection*{SHI.2.2.2 General form}
\label{SHI:asympt:0:genf}



\begin{equation*}
\label{SHI:asympt:0:genfsum}
\begin{split}
& \operatorname{Shi} (x)\approx x \sum_{n = 0}^{\infty} u (n) x^{n}.
\end{split}\tag{SHI.2.2.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{SHI: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{SHI.2.2.2.2}
\end{equation*}
Initial conditions of SHI.2.2.2.2 are given by
\begin{equation*}
\label{SHI:asympt:0:genfic}
\begin{split}
u (1)& =0, \\
u (0)& =1, \\
u (2)& =\frac{1}{18}.
\end{split}\tag{SHI.2.2.2.3}
\end{equation*}
The recurrence SHI.2.2.2.2 has the closed form solution
\begin{equation*}
\label{SHI:asympt:0:RDINREFRDGENFROMRDCLOSED}
\begin{split}
u (2 n + 1)& =0, \\
u (2 n)& =\frac{1}{(2 n + 1) \Gamma (2 n + 2)}.
\end{split}\tag{SHI.2.2.2.4}
\end{equation*}

\end{document}