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\begin{document}
\chapter*{HASC Inverse Hyperbolic Secant}
\label{HASC}
\section*{HASC.1 Introduction}
\label{HASC:intro}


Let $x$ be a complex variable of $\mathbb{C} \setminus \{0\}$.The function Inverse Hyperbolic Secant (noted $\operatorname{arcsech}$) is defined by the following second order differential equation
\begin{equation*}
\label{HASC: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{HASC.1.1}
\end{equation*}



The initial conditions of HASC.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/ASC/ASC.html#ASC}{Inverse Secant},\href{http://algo.inria.fr/esf/function/ACCS/ACCS.html#ACCS}{Inverse Cosecant}
\section*{HASC.2 Series and asymptotic expansions}
\label{HASC:asympt}
\subsection*{HASC.2.1 Asymptotic expansion at $1$}
\label{74359523722028997}
\subsubsection*{HASC.2.1.1 First terms}
\label{HASC:asympt:1:termsec}



\begin{equation*}
\label{HASC:asympt:1:terms}
\begin{split}
& \operatorname{arcsech} (x)\approx \sqrt{x - 1} \biggl(\frac{47442055 i}{637534208} \sqrt{2} (x - 1)^{9} - i \sqrt{2} -  \\
& \quad{}\quad{}\frac{24295375159 i}{429496729600} \sqrt{2} (x - 1)^{12} + \frac{3109879375897 i}{68169720922112} \sqrt{2} (x - 1)^{15} -  \\
& \quad{}\quad{}\frac{43 i}{160} \sqrt{2} (x - 1)^{2} + \frac{5 i}{12} \sqrt{2} (x - 1) + \frac{74069 i}{786432} \sqrt{2} (x - 1)^{7} +  \\
& \quad{}\quad{}\frac{1518418695 i}{24696061952} \sqrt{2} (x - 1)^{11} - \frac{92479 i}{851968} \sqrt{2} (x - 1)^{6} -  \\
& \quad{}\quad{}\frac{126527543 i}{1879048192} \sqrt{2} (x - 1)^{10} - \frac{2867 i}{18432} \sqrt{2} (x - 1)^{4} -  \\
& \quad{}\quad{}\frac{11857475 i}{142606336} \sqrt{2} (x - 1)^{8} + \frac{177 i}{896} \sqrt{2} (x - 1)^{3} +  \\
& \quad{}\quad{}\frac{11531 i}{90112} \sqrt{2} (x - 1)^{5} - \frac{777467420263 i}{15942918602752} \sqrt{2} (x - 1)^{14} +  \\
& \quad{}\quad{}\frac{97182800711 i}{1855425871872} \sqrt{2} (x - 1)^{13}\ldots\biggr).
\end{split}\tag{HASC.2.1.1.1}
\end{equation*}

\subsubsection*{HASC.2.1.2 General form}
\label{HASC:asympt:1:genf}



\begin{equation*}
\label{HASC:asympt:1:genfsum}
\begin{split}
& \operatorname{arcsech} (x)\approx \sqrt{x - 1} \sum_{n = 0}^{\infty} u (n) (x - 1)^{n}.
\end{split}\tag{HASC.2.1.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{HASC: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{HASC.2.1.2.2}
\end{equation*}
Initial conditions of HASC.2.1.2.2 are given by
\begin{equation*}
\label{HASC:asympt:1:genfic}
\begin{split}
u (0)& =-i\sqrt{2}, \\
u (1)& =\frac{5 i}{12} \sqrt{2}.
\end{split}\tag{HASC.2.1.2.3}
\end{equation*}

\subsection*{HASC.2.2 Asymptotic expansion at $-1$}
\label{74359581504119137}
\subsubsection*{HASC.2.2.1 First terms}
\label{HASC:asympt:TB1:termsec}



\begin{equation*}
\label{HASC:asympt:TB1:terms}
\begin{split}
& \operatorname{arcsech} (x)\approx (i \pi\ldots) + \sqrt{x + 1} \Biggl(-\sqrt{2} - \frac{5 (x + 1) \sqrt{2}}{12} - \frac{43 (x + 1)^{2} \sqrt{2}}{160} -  \\
& \quad{}\quad{}\frac{177 (x + 1)^{3} \sqrt{2}}{896} - \frac{2867 (x + 1)^{4} \sqrt{2}}{18432} - \frac{11531 (x + 1)^{5} \sqrt{2}}{90112} -  \\
& \quad{}\quad{}\frac{92479 (x + 1)^{6} \sqrt{2}}{851968} - \frac{74069 (x + 1)^{7} \sqrt{2}}{786432} -  \\
& \quad{}\quad{}\frac{11857475 (x + 1)^{8} \sqrt{2}}{142606336}\ldots\Biggr).
\end{split}\tag{HASC.2.2.1.1}
\end{equation*}

\subsubsection*{HASC.2.2.2 General form}
\label{743595939158221370}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{HASC.2.3 Asymptotic expansion at $0$}
\label{743593477712506824}
\subsubsection*{HASC.2.3.1 First terms}
\label{HASC:asympt:0:termsec}



\begin{equation*}
\label{HASC:asympt:0:terms}
\begin{split}
& \operatorname{arcsech} (x)\approx \Biggl(\frac{x^{2}}{4} + \frac{3 x^{4}}{32} + \frac{5 x^{6}}{96} + \frac{35 x^{8}}{1024} + \operatorname{ln} (x) + \operatorname{ln} (2)\ldots\Biggr).
\end{split}\tag{HASC.2.3.1.1}
\end{equation*}

\subsubsection*{HASC.2.3.2 General form}
\label{743593381020411035}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\end{document}