\documentclass[]{amsbook}
\usepackage{amsmath,amsopn}
\usepackage{graphicx}
\usepackage{hyperref,url}
% BEGIN SPECIAL COMMANDS
\newcommand{\MADerror}[1]{\begin{flushleft}\fbox{\begin{minipage}{\textwidth}{\tt #1}\end{minipage}}\end{flushleft}}
\newcommand{\MADierror}[1]{\fbox{\tt #1}}
% END SPECIAL COMMANDS
% BEGIN STATIC HEADER

% END STATIC HEADER
% BEGIN DYNAMIC HEADER
% Copyright \copyright 2001-2003 by the Algorithms Project and INRIA. All rigths reserved.

% END DYNAMIC HEADER
\begin{document}
\chapter*{HASN Inverse Hyperbolic Sine}
\label{HASN}
\section*{HASN.1 Introduction}
\label{HASN:intro}


Let $x$ be a complex variable of $\mathbb{C} \setminus \{\infty\}$.The function Inverse Hyperbolic Sine (noted $\operatorname{arcsinh}$) is defined by the following second order differential equation
\begin{equation*}
\label{HASN:diffeq}
\begin{split}
x \frac{\partial y (x)}{\partial x} + \bigl(1 + x^{2}\bigr) \frac{\partial^{2} y (x)}{\partial x^{2}}& =0.
\end{split}\tag{HASN.1.1}
\end{equation*}



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

\section*{HASN.2 Series and asymptotic expansions}
\label{HASN:asympt}
\subsection*{HASN.2.1 Asymptotic expansion at $-i$}
\label{74359774567110100}
\subsubsection*{HASN.2.1.1 First terms}
\label{HASN:asympt:TBI:termsec}



\begin{equation*}
\label{HASN:asympt:TBI:terms}
\begin{split}
& \operatorname{arcsinh} (x)\approx \biggl(-\frac{i}{2}\pi\ldots\biggr) + \sqrt{x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)} \Biggl(1 + i -  \\
& \quad{}\quad{}\frac{(1 + i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)}{12 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)} +  \\
& \quad{}\quad{}\frac{(3 + 3 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{2}}{160 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}} -  \\
& \quad{}\quad{}\frac{(5 + 5 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{3}}{896 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{3}} +  \\
& \quad{}\quad{}\frac{(35 + 35 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{4}}{18432 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{4}} -  \\
& \quad{}\quad{}\frac{(63 + 63 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{5}}{90112 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{5}} +  \\
& \quad{}\quad{}\frac{(231 + 231 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{6}}{851968 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{6}} -  \\
& \quad{}\quad{}\frac{(143 + 143 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{7}}{1310720 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{7}} +  \\
& \quad{}\quad{}\frac{(6435 + 6435 i) \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{8}}{142606336 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{8}}\ldots\Biggr).
\end{split}\tag{HASN.2.1.1.1}
\end{equation*}

\subsubsection*{HASN.2.1.2 General form}
\label{743597149995539677}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{HASN.2.2 Asymptotic expansion at $\infty$}
\label{74359752630297103}
\subsubsection*{HASN.2.2.1 First terms}
\label{HASN:asympt:infinity:termsec}



\begin{equation*}
\label{HASN:asympt:infinity:terms}
\begin{split}
& \operatorname{arcsinh} (x)\approx \Biggl(\operatorname{ln} (2) - \frac{1}{4 x^{2}} + \frac{3}{32 x^{4}} - \frac{5}{96 x^{6}} + \frac{35}{1024 x^{8}} + \operatorname{ln} \biggl(\frac{1}{x}\biggr)\ldots\Biggr).
\end{split}\tag{HASN.2.2.1.1}
\end{equation*}

\subsubsection*{HASN.2.2.2 General form}
\label{743597647474017340}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{HASN.2.3 Taylor expansion at $0$}
\label{743597999696755868}
\subsubsection*{HASN.2.3.1 First terms}
\label{743597849759450576}



\begin{equation*}
\label{HASN:asympt:0:RDLBLRDTERMSRDEQ}
\begin{split}
\operatorname{arcsinh} (x)& =x - \frac{1}{6}   x^{3} + \frac{3}{40}   x^{5} - \frac{5}{112}   x^{7} + \frac{35}{1152}   x^{9} - \frac{63}{2816}   x^{11} + \frac{231}{13312}    \\
& \quad{}\quad{}x^{13} - \frac{143}{10240}   x^{15} + \operatorname{O} \bigl(x^{16}\bigr).
\end{split}\tag{HASN.2.3.1.1}
\end{equation*}

\subsubsection*{HASN.2.3.2 General form}
\label{743597494990109413}



\begin{equation*}
\label{HASN:asympt:0:RDLBLRDGENFORMRDEQ}
\begin{split}
\operatorname{arcsinh} (x)& =\sum_{n = 0}^{\infty} u (n) x^{n}.
\end{split}\tag{HASN.2.3.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{HASN:asympt:0:toto}
\begin{split}
n^{2} u (n) + \bigl(n^{2} + 3 n + 2\bigr) u (n + 2)& =0.
\end{split}\tag{HASN.2.3.2.2}
\end{equation*}
Initial conditions of HASN.2.3.2.2 are given by
\begin{equation*}
\label{HASN:asympt:0:RDLBLRDGENFORMRDIC}
\begin{split}
u (0)& =0, \\
u (1)& =1.
\end{split}\tag{HASN.2.3.2.3}
\end{equation*}

\subsection*{HASN.2.4 Asymptotic expansion at $i$}
\label{743597614806287609}
\subsubsection*{HASN.2.4.1 First terms}
\label{HASN:asympt:I:termsec}



\begin{equation*}
\label{HASN:asympt:I:terms}
\begin{split}
& \operatorname{arcsinh} (x)\approx \biggl(\frac{i}{2} \pi\ldots\biggr) + \sqrt{x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)} \Biggl(1 - i -  \\
& \quad{}\quad{}\frac{(1 - i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)}{12 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)} +  \\
& \quad{}\quad{}\frac{(3 - 3 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{2}}{160 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}} -  \\
& \quad{}\quad{}\frac{(5 - 5 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{3}}{896 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{3}} +  \\
& \quad{}\quad{}\frac{(35 - 35 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{4}}{18432 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{4}} -  \\
& \quad{}\quad{}\frac{(63 - 63 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{5}}{90112 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{5}} +  \\
& \quad{}\quad{}\frac{(231 - 231 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{6}}{851968 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{6}} -  \\
& \quad{}\quad{}\frac{(143 - 143 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{7}}{1310720 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{7}} +  \\
& \quad{}\quad{}\frac{(6435 - 6435 i) \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{8}}{142606336 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{8}}\ldots\Biggr).
\end{split}\tag{HASN.2.4.1.1}
\end{equation*}

\subsubsection*{HASN.2.4.2 General form}
\label{743597619078762853}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\section*{HASN.3 Graphs}
\label{743605289145218184}
\subsection*{HASN.3.1 Real axis}
\label{743605646420438224}
\begin{center}
\includegraphics[width=6cm]{HASN/744629903902225771}
\end{center}
\subsection*{HASN.3.2 Complex plane}
\label{743605936304259889}
\begin{center}
\includegraphics[width=6cm]{HASN/744629869971578915}
\end{center}
\end{document}