\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*{ASC Inverse Secant}
\label{ASC}
\section*{ASC.1 Introduction}
\label{ASC:intro}


Let $x$ be a complex variable of $\mathbb{C} \setminus \{0\}$.The function Inverse Secant (noted $\operatorname{arcsec}$) is defined by the following second order differential equation
\begin{equation*}
\label{ASC:diffeq}
\begin{split}
\bigl(2 x^{2} - 1\bigr) \frac{\partial y (x)}{\partial x} + \bigl(x^{3} - x\bigr) \frac{\partial^{2} y (x)}{\partial x^{2}}& =0.
\end{split}\tag{ASC.1.1}
\end{equation*}



The initial conditions of ASC.1.1 at $0$ are not simple to state, since $0$ is a (regular) singular point.


Related functions: \href{http://algo.inria.fr/esf/function/HASC/HASC.html#HASC}{Inverse Hyperbolic Secant},\href{http://algo.inria.fr/esf/function/ACCS/ACCS.html#ACCS}{Inverse Cosecant}
\section*{ASC.2 Series and asymptotic expansions}
\label{ASC:asympt}
\subsection*{ASC.2.1 Asymptotic expansion at $-1$}
\label{74330567060541266}
\subsubsection*{ASC.2.1.1 First terms}
\label{ASC:asympt:TB1:termsec}



\begin{equation*}
\label{ASC:asympt:TB1:terms}
\begin{split}
& \operatorname{arcsec} (x)\approx (\pi\ldots) + \sqrt{x + 1} \biggl(-i\sqrt{2} - \frac{5 i}{12} (x + 1) \sqrt{2} - \frac{43 i}{160} (x + 1)^{2} \sqrt{2}  \\
& \quad{}\quad{}- \frac{177 i}{896} (x + 1)^{3} \sqrt{2} - \frac{2867 i}{18432} (x + 1)^{4} \sqrt{2} -  \\
& \quad{}\quad{}\frac{11531 i}{90112} (x + 1)^{5} \sqrt{2} - \frac{92479 i}{851968} (x + 1)^{6} \sqrt{2} -  \\
& \quad{}\quad{}\frac{74069 i}{786432} (x + 1)^{7} \sqrt{2} - \frac{11857475 i}{142606336} (x + 1)^{8} \sqrt{2}\ldots\biggr).
\end{split}\tag{ASC.2.1.1.1}
\end{equation*}

\subsubsection*{ASC.2.1.2 General form}
\label{74330584125842236}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{ASC.2.2 Asymptotic expansion at $0$}
\label{743303898724880795}
\subsubsection*{ASC.2.2.1 First terms}
\label{ASC:asympt:0:termsec}



\begin{equation*}
\label{ASC:asympt:0:terms}
\begin{split}
& \operatorname{arcsec} (x)\approx  \\
& \quad{}\quad{}\biggl(i \operatorname{ln} (2) + \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} (x)\ldots\biggr).
\end{split}\tag{ASC.2.2.1.1}
\end{equation*}

\subsubsection*{ASC.2.2.2 General form}
\label{743303433599229456}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{ASC.2.3 Asymptotic expansion at $1$}
\label{743305549552888716}
\subsubsection*{ASC.2.3.1 First terms}
\label{ASC:asympt:1:termsec}



\begin{equation*}
\label{ASC:asympt:1:terms}
\begin{split}
& \operatorname{arcsec} (x)\approx \sqrt{x - 1} \Biggl(\sqrt{2} - \frac{5 \sqrt{2} (x - 1)}{12} + \frac{43 \sqrt{2} (x - 1)^{2}}{160} -  \\
& \quad{}\quad{}\frac{177 \sqrt{2} (x - 1)^{3}}{896} + \frac{2867 \sqrt{2} (x - 1)^{4}}{18432} - \frac{11531 \sqrt{2} (x - 1)^{5}}{90112} +  \\
& \quad{}\quad{}\frac{92479 \sqrt{2} (x - 1)^{6}}{851968} - \frac{74069 \sqrt{2} (x - 1)^{7}}{786432} +  \\
& \quad{}\quad{}\frac{11857475 \sqrt{2} (x - 1)^{8}}{142606336} - \frac{47442055 \sqrt{2} (x - 1)^{9}}{637534208} +  \\
& \quad{}\quad{}\frac{126527543 \sqrt{2} (x - 1)^{10}}{1879048192} - \frac{1518418695 \sqrt{2} (x - 1)^{11}}{24696061952} +  \\
& \quad{}\quad{}\frac{24295375159 \sqrt{2} (x - 1)^{12}}{429496729600} - \frac{97182800711 \sqrt{2} (x - 1)^{13}}{1855425871872} +  \\
& \quad{}\quad{}\frac{777467420263 \sqrt{2} (x - 1)^{14}}{15942918602752} - \frac{3109879375897 \sqrt{2} (x - 1)^{15}}{68169720922112}\ldots\Biggr).
\end{split}\tag{ASC.2.3.1.1}
\end{equation*}

\subsubsection*{ASC.2.3.2 General form}
\label{ASC:asympt:1:genf}



\begin{equation*}
\label{ASC:asympt:1:genfsum}
\begin{split}
& \operatorname{arcsec} (x)\approx \sqrt{x - 1} \sum_{n = 0}^{\infty} u (n) (x - 1)^{n}.
\end{split}\tag{ASC.2.3.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{ASC:asympt:1:genfrec}
\begin{split}
& 2 u (n) \biggl(\frac{1}{2} + n\biggr) n + u (n - 1) \biggl(-\frac{1}{2} + n\biggr) \biggl(-\frac{1}{2} + 3 n\biggr) + u (n - 2) \biggl(-\frac{3}{2} + n\biggr) \biggl(-\frac{1}{2} + n\biggr) \\
& \quad{}\quad{}=0.
\end{split}\tag{ASC.2.3.2.2}
\end{equation*}
Initial conditions of ASC.2.3.2.2 are given by
\begin{equation*}
\label{ASC:asympt:1:genfic}
\begin{split}
u (0)& =\sqrt{2}, \\
u (1)& =\frac{-5\sqrt{2}}{12}.
\end{split}\tag{ASC.2.3.2.3}
\end{equation*}

\end{document}