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


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



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



Related function: \href{http://algo.inria.fr/esf/function/HACT/HACT.html#HACT}{Inverse Hyperbolic Cotangent}
\section*{HATN.2 Series and asymptotic expansions}
\label{HATN:asympt}
\subsection*{HATN.2.1 Taylor expansion at $0$}
\label{743606696514914779}
\subsubsection*{HATN.2.1.1 First terms}
\label{74360622895394885}



\begin{equation*}
\label{HATN:asympt:0:RDLBLRDTERMSRDEQ}
\begin{split}
\operatorname{arctanh} (x)& =x + \frac{1}{3}   x^{3} + \frac{1}{5}   x^{5} + \frac{1}{7}   x^{7} + \frac{1}{9}   x^{9} + \frac{1}{11}   x^{11} + \frac{1}{13}   x^{13} + \frac{1}{15}   x^{15}  \\
& \quad{}\quad{}+ \operatorname{O} \bigl(x^{16}\bigr).
\end{split}\tag{HATN.2.1.1.1}
\end{equation*}

\subsubsection*{HATN.2.1.2 General form}
\label{743606534717884810}



\begin{equation*}
\label{HATN:asympt:0:RDLBLRDGENFORMRDEQ}
\begin{split}
\operatorname{arctanh} (x)& =\sum_{n = 0}^{\infty} u (n) x^{n}.
\end{split}\tag{HATN.2.1.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{HATN:asympt:0:toto}
\begin{split}
n u (n) - -(-n - 2) u (n + 2)& =0.
\end{split}\tag{HATN.2.1.2.2}
\end{equation*}
Initial conditions of HATN.2.1.2.2 are given by
\begin{equation*}
\label{HATN:asympt:0:RDLBLRDGENFORMRDIC}
\begin{split}
u (1)& =1, \\
u (0)& =0.
\end{split}\tag{HATN.2.1.2.3}
\end{equation*}

\subsection*{HATN.2.2 Asymptotic expansion at $1$}
\label{74360639901096090}
\subsubsection*{HATN.2.2.1 First terms}
\label{HATN:asympt:1:termsec}



\begin{equation*}
\label{HATN:asympt:1:terms}
\begin{split}
& \operatorname{arctanh} (x)\approx \Biggl(\frac{i}{2} \pi + \frac{\operatorname{ln} (2)}{2} - \frac{x}{4} + \frac{1}{4} + \frac{(x - 1)^{2}}{16} - \frac{(x - 1)^{3}}{48} + \frac{(x - 1)^{4}}{128} -  \\
& \quad{}\quad{}\frac{(x - 1)^{5}}{320} + \frac{(x - 1)^{6}}{768} - \frac{(x - 1)^{7}}{1792} + \frac{(x - 1)^{8}}{4096} + \frac{\operatorname{ln} (x - 1)}{2}\ldots\Biggr).
\end{split}\tag{HATN.2.2.1.1}
\end{equation*}

\subsubsection*{HATN.2.2.2 General form}
\label{74360695842166367}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{HATN.2.3 Asymptotic expansion at $-1$}
\label{743606833725669197}
\subsubsection*{HATN.2.3.1 First terms}
\label{HATN:asympt:TB1:termsec}



\begin{equation*}
\label{HATN:asympt:TB1:terms}
\begin{split}
& \operatorname{arctanh} (x)\approx \Biggl(\frac{-\operatorname{ln} (2)}{2} - \frac{x}{4} - \frac{1}{4} - \frac{(x + 1)^{2}}{16} - \frac{(x + 1)^{3}}{48} - \frac{(x + 1)^{4}}{128} - \frac{(x + 1)^{5}}{320}  \\
& \quad{}\quad{}- \frac{(x + 1)^{6}}{768} - \frac{(x + 1)^{7}}{1792} - \frac{(x + 1)^{8}}{4096} - \frac{\operatorname{ln} (x + 1)}{2}\ldots\Biggr).
\end{split}\tag{HATN.2.3.1.1}
\end{equation*}

\subsubsection*{HATN.2.3.2 General form}
\label{743606944582639436}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\section*{HATN.3 Graphs}
\label{743612404796668928}
\subsection*{HATN.3.1 Real axis}
\label{743612436873243953}
\begin{center}
\includegraphics[width=6cm]{HATN/744651743267339661}
\end{center}
\subsection*{HATN.3.2 Complex plane}
\label{743612788254708175}
\begin{center}
\includegraphics[width=6cm]{HATN/744651114404605501}
\end{center}
\end{document}