\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*{ASN Inverse Sine}
\label{ASN}
\section*{ASN.1 Introduction}
\label{ASN:intro}


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



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



Related functions: \href{http://algo.inria.fr/esf/function/ACS/ACS.html#ACS}{Inverse Cosine},\href{http://algo.inria.fr/esf/function/HACS/HACS.html#HACS}{Inverse Hyperbolic Cosine}
\section*{ASN.2 Series and asymptotic expansions}
\label{ASN:asympt}
\subsection*{ASN.2.1 Asymptotic expansion at $-1$}
\label{743305810557140299}
\subsubsection*{ASN.2.1.1 First terms}
\label{ASN:asympt:TB1:termsec}



\begin{equation*}
\label{ASN:asympt:TB1:terms}
\begin{split}
& \operatorname{arcsin} (x)\approx \biggl(\frac{-\pi}{2}\ldots\biggr) + \sqrt{x + 1} \Biggl(\sqrt{2} + \frac{(x + 1) \sqrt{2}}{12} + \frac{3 (x + 1)^{2} \sqrt{2}}{160} +  \\
& \quad{}\quad{}\frac{5 (x + 1)^{3} \sqrt{2}}{896} + \frac{35 (x + 1)^{4} \sqrt{2}}{18432} + \frac{63 (x + 1)^{5} \sqrt{2}}{90112} +  \\
& \quad{}\quad{}\frac{231 (x + 1)^{6} \sqrt{2}}{851968} + \frac{143 (x + 1)^{7} \sqrt{2}}{1310720} + \frac{6435 (x + 1)^{8} \sqrt{2}}{142606336}\ldots\Biggr).
\end{split}\tag{ASN.2.1.1.1}
\end{equation*}

\subsubsection*{ASN.2.1.2 General form}
\label{743305985479921284}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{ASN.2.2 Asymptotic expansion at $\infty$}
\label{743306885805483826}
\subsubsection*{ASN.2.2.1 First terms}
\label{ASN:asympt:infinity:termsec}



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

\subsubsection*{ASN.2.2.2 General form}
\label{74330659712272244}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{ASN.2.3 Taylor expansion at $0$}
\label{743305568046682595}
\subsubsection*{ASN.2.3.1 First terms}
\label{743305887974857856}



\begin{equation*}
\label{ASN:asympt:0:RDLBLRDTERMSRDEQ}
\begin{split}
\operatorname{arcsin} (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{ASN.2.3.1.1}
\end{equation*}

\subsubsection*{ASN.2.3.2 General form}
\label{743305788553867838}



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

\subsection*{ASN.2.4 Asymptotic expansion at $1$}
\label{743305223609305579}
\subsubsection*{ASN.2.4.1 First terms}
\label{ASN:asympt:1:termsec}



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

\subsubsection*{ASN.2.4.2 General form}
\label{743305651198106710}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\section*{ASN.3 Graphs}
\label{743311134912196910}
\subsection*{ASN.3.1 Real axis}
\label{743311208735338387}
\begin{center}
\includegraphics[width=6cm]{ASN/743885198008607357}
\end{center}
\subsection*{ASN.3.2 Complex plane}
\label{743311863489794196}
\begin{center}
\includegraphics[width=6cm]{ASN/743885649562536669}
\end{center}
\end{document}