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


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



The initial conditions of HACS.1.1 are given at $0$ by
\begin{equation*}
\label{HACS:inicond}
\begin{split}
\operatorname{arccosh} (0)& =\frac{i}{2} \pi, \\
\frac{\partial \operatorname{arccosh} (x)}{\partial x} (0)& =-i.
\end{split}\tag{HACS.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/ASN/ASN.html#ASN}{Inverse Sine}
\section*{HACS.2 Series and asymptotic expansions}
\label{HACS:asympt}
\subsection*{HACS.2.1 Asymptotic expansion at $-1$}
\label{743575536763019788}
\subsubsection*{HACS.2.1.1 First terms}
\label{HACS:asympt:TB1:termsec}



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

\subsubsection*{HACS.2.1.2 General form}
\label{743575543435760000}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{HACS.2.2 Asymptotic expansion at $1$}
\label{743575711155171933}
\subsubsection*{HACS.2.2.1 First terms}
\label{HACS:asympt:1:termsec}



\begin{equation*}
\label{HACS:asympt:1:terms}
\begin{split}
& \operatorname{arccosh} (x)\approx \sqrt{x - 1} \Biggl(\sqrt{2} - \frac{\sqrt{2} (x - 1)}{12} + \frac{3 \sqrt{2} (x - 1)^{2}}{160} -  \\
& \quad{}\quad{}\frac{5 \sqrt{2} (x - 1)^{3}}{896} + \frac{35 \sqrt{2} (x - 1)^{4}}{18432} - \frac{63 \sqrt{2} (x - 1)^{5}}{90112} +  \\
& \quad{}\quad{}\frac{231 \sqrt{2} (x - 1)^{6}}{851968} - \frac{143 \sqrt{2} (x - 1)^{7}}{1310720} + \frac{6435 \sqrt{2} (x - 1)^{8}}{142606336} -  \\
& \quad{}\quad{}\frac{12155 \sqrt{2} (x - 1)^{9}}{637534208} + \frac{46189 \sqrt{2} (x - 1)^{10}}{5637144576} - \frac{88179 \sqrt{2} (x - 1)^{11}}{24696061952}  \\
& \quad{}\quad{}+ \frac{676039 \sqrt{2} (x - 1)^{12}}{429496729600} - \frac{1300075 \sqrt{2} (x - 1)^{13}}{1855425871872} +  \\
& \quad{}\quad{}\frac{5014575 \sqrt{2} (x - 1)^{14}}{15942918602752} - \frac{9694845 \sqrt{2} (x - 1)^{15}}{68169720922112}\ldots\Biggr).
\end{split}\tag{HACS.2.2.1.1}
\end{equation*}

\subsubsection*{HACS.2.2.2 General form}
\label{HACS:asympt:1:genf}



\begin{equation*}
\label{HACS:asympt:1:genfsum}
\begin{split}
& \operatorname{arccosh} (x)\approx \sqrt{x - 1} \sum_{n = 0}^{\infty} u (n) (x - 1)^{n}.
\end{split}\tag{HACS.2.2.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{HACS:asympt:1:genfrec}
\begin{split}
2 u (n) \biggl(n + \frac{1}{2}\biggr) n + u (n - 1) \biggl(-\frac{1}{2} + n\biggr)^{2}& =0.
\end{split}\tag{HACS.2.2.2.2}
\end{equation*}
Initial conditions of HACS.2.2.2.2 are given by
\begin{equation*}
\label{HACS:asympt:1:genfic}
\begin{split}
u (0)& =\sqrt{2}.
\end{split}\tag{HACS.2.2.2.3}
\end{equation*}
The recurrence HACS.2.2.2.2 has the closed form solution
\begin{equation*}
\label{HACS:asympt:1:RDINREFRDGENFROMRDCLOSED}
\begin{split}
u (n)& =\frac{2^{\bigl(n + \frac{1}{2}\bigr)} \Gamma \Bigl(n + \frac{1}{2}\Bigr) (-1)^{n}}{4^{n} \sqrt{\pi} \Gamma (n + 1) (2 n + 1)}.
\end{split}\tag{HACS.2.2.2.4}
\end{equation*}

\subsection*{HACS.2.3 Asymptotic expansion at $\infty$}
\label{743575463630824034}
\subsubsection*{HACS.2.3.1 First terms}
\label{HACS:asympt:infinity:termsec}



\begin{equation*}
\label{HACS:asympt:infinity:terms}
\begin{split}
& \operatorname{arccosh} (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{HACS.2.3.1.1}
\end{equation*}

\subsubsection*{HACS.2.3.2 General form}
\label{743575188097894389}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{HACS.2.4 Taylor expansion at $0$}
\label{743574653375632101}
\subsubsection*{HACS.2.4.1 First terms}
\label{743574539301373623}



\begin{equation*}
\label{HACS:asympt:0:RDLBLRDTERMSRDEQ}
\begin{split}
\operatorname{arccosh} (x)& =\frac{i}{2} \pi - i   x - \frac{i}{6}   x^{3} - \frac{3 i}{40}   x^{5} - \frac{5 i}{112}   x^{7} - \frac{35 i}{1152}   x^{9} - \frac{63 i}{2816}   x^{11} -  \\
& \quad{}\quad{}\frac{231 i}{13312}   x^{13} - \frac{143 i}{10240}   x^{15} + \operatorname{O} \bigl(x^{16}\bigr).
\end{split}\tag{HACS.2.4.1.1}
\end{equation*}

\subsubsection*{HACS.2.4.2 General form}
\label{74357436390268487}



\begin{equation*}
\label{HACS:asympt:0:RDLBLRDGENFORMRDEQ}
\begin{split}
\operatorname{arccosh} (x)& =\sum_{n = 0}^{\infty} u (n) x^{n}.
\end{split}\tag{HACS.2.4.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{HACS:asympt:0:toto}
\begin{split}
n^{2} u (n) - -\bigl(-n^{2} - 3 n - 2\bigr) u (n + 2)& =0.
\end{split}\tag{HACS.2.4.2.2}
\end{equation*}
Initial conditions of HACS.2.4.2.2 are given by
\begin{equation*}
\label{HACS:asympt:0:RDLBLRDGENFORMRDIC}
\begin{split}
u (0)& =\frac{i}{2} \pi, \\
u (1)& =-i.
\end{split}\tag{HACS.2.4.2.3}
\end{equation*}

\section*{HACS.3 Graphs}
\label{743583663018105954}
\subsection*{HACS.3.1 Real axis}
\label{743583228868570073}
\begin{center}
\includegraphics[width=6cm]{HACS/744279492159209078}
\end{center}
\subsection*{HACS.3.2 Complex plane}
\label{743583719372226237}
\begin{center}
\includegraphics[width=6cm]{HACS/744279599394906409}
\end{center}
\end{document}