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\begin{document}
\chapter*{ACCS Inverse Cosecant}
\label{ACCS}
\section*{ACCS.1 Introduction}
\label{ACCS:intro}


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



The initial conditions of ACCS.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/ASC/ASC.html#ASC}{Inverse Secant}
\section*{ACCS.2 Series and asymptotic expansions}
\label{ACCS:asympt}
\subsection*{ACCS.2.1 Asymptotic expansion at $-1$}
\label{743276604305613921}
\subsubsection*{ACCS.2.1.1 First terms}
\label{ACCS:asympt:TB1:termsec}



\begin{equation*}
\label{ACCS:asympt:TB1:terms}
\begin{split}
& \operatorname{arccsc} (x)\approx \biggl(\frac{-\pi}{2}\ldots\biggr) + \sqrt{x + 1} \biggl(i \sqrt{2} + \frac{5 i}{12} (x + 1) \sqrt{2} +  \\
& \quad{}\quad{}\frac{43 i}{160} (x + 1)^{2} \sqrt{2} + \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{ACCS.2.1.1.1}
\end{equation*}

\subsubsection*{ACCS.2.1.2 General form}
\label{743276722974121768}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{ACCS.2.2 Asymptotic expansion at $0$}
\label{743276558458718976}
\subsubsection*{ACCS.2.2.1 First terms}
\label{ACCS:asympt:0:termsec}



\begin{equation*}
\label{ACCS:asympt:0:terms}
\begin{split}
& \operatorname{arccsc} (x)\approx \biggl( \\
& \quad{}\quad{}-i\operatorname{ln} (2) + \frac{\pi}{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{ACCS.2.2.1.1}
\end{equation*}

\subsubsection*{ACCS.2.2.2 General form}
\label{743276474256143563}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{ACCS.2.3 Asymptotic expansion at $1$}
\label{74327632062222085}
\subsubsection*{ACCS.2.3.1 First terms}
\label{ACCS:asympt:1:termsec}



\begin{equation*}
\label{ACCS:asympt:1:terms}
\begin{split}
& \operatorname{arccsc} (x)\approx \biggl(\frac{\pi}{2}\ldots\biggr) + \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{ACCS.2.3.1.1}
\end{equation*}

\subsubsection*{ACCS.2.3.2 General form}
\label{743276746753830538}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\end{document}