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


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



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

\section*{ATN.2 Series and asymptotic expansions}
\label{ATN:asympt}
\subsection*{ATN.2.1 Asymptotic expansion at $i$}
\label{743312425237383431}
\subsubsection*{ATN.2.1.1 First terms}
\label{ATN:asympt:I:termsec}



\begin{equation*}
\label{ATN:asympt:I:terms}
\begin{split}
& \operatorname{arctan} (x)\approx \Biggl(\frac{i}{2} \operatorname{ln} (2) + \frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr) \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)}{4}  \\
& \quad{}\quad{}- \frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{2}}{16} + \frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{3}}{48 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)} -  \\
& \quad{}\quad{}\frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{4}}{128 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}} + \frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{5}}{320 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{3}} -  \\
& \quad{}\quad{}\frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{6}}{768 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{4}} + \frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{7}}{1792 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{5}} -  \\
& \quad{}\quad{}\frac{\bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)^{8}}{4096 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{6}} + \frac{\operatorname{ln} \bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\bigr)}{2}\ldots\Biggr).
\end{split}\tag{ATN.2.1.1.1}
\end{equation*}

\subsubsection*{ATN.2.1.2 General form}
\label{743312852497310816}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{ATN.2.2 Asymptotic expansion at $-i$}
\label{743313351252190359}
\subsubsection*{ATN.2.2.1 First terms}
\label{ATN:asympt:TBI:termsec}



\begin{equation*}
\label{ATN:asympt:TBI:terms}
\begin{split}
& \operatorname{arctan} (x)\approx \Biggl(-\frac{i}{2}\operatorname{ln} (2) + \frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr) \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)}{4}  \\
& \quad{}\quad{}- \frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{2}}{16} + \frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{3}}{48 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)} -  \\
& \quad{}\quad{}\frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{4}}{128 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}} + \frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{5}}{320 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{3}} -  \\
& \quad{}\quad{}\frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{6}}{768 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{4}} + \frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{7}}{1792 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{5}} -  \\
& \quad{}\quad{}\frac{\bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)^{8}}{4096 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{6}} + \frac{\operatorname{ln} \bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\bigr)}{2}\ldots\Biggr).
\end{split}\tag{ATN.2.2.1.1}
\end{equation*}

\subsubsection*{ATN.2.2.2 General form}
\label{743313947635435699}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{ATN.2.3 Taylor expansion at $0$}
\label{743313521410145117}
\subsubsection*{ATN.2.3.1 First terms}
\label{743313194640455210}



\begin{equation*}
\label{ATN:asympt:0:RDLBLRDTERMSRDEQ}
\begin{split}
\operatorname{arctan} (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{ATN.2.3.1.1}
\end{equation*}

\subsubsection*{ATN.2.3.2 General form}
\label{743313737161973361}



\begin{equation*}
\label{ATN:asympt:0:RDLBLRDGENFORMRDEQ}
\begin{split}
\operatorname{arctan} (x)& =\sum_{n = 0}^{\infty} u (n) x^{n}.
\end{split}\tag{ATN.2.3.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{ATN:asympt:0:toto}
\begin{split}
-n u (n) - -(-n - 2) u (n + 2)& =0.
\end{split}\tag{ATN.2.3.2.2}
\end{equation*}
Initial conditions of ATN.2.3.2.2 are given by
\begin{equation*}
\label{ATN:asympt:0:RDLBLRDGENFORMRDIC}
\begin{split}
u (0)& =0, \\
u (1)& =1.
\end{split}\tag{ATN.2.3.2.3}
\end{equation*}

\section*{ATN.3 Graphs}
\label{743318324437166624}
\subsection*{ATN.3.1 Real axis}
\label{743318835798495990}
\begin{center}
\includegraphics[width=6cm]{ATN/743909961058614330}
\end{center}
\subsection*{ATN.3.2 Complex plane}
\label{743318372909367548}
\begin{center}
\includegraphics[width=6cm]{ATN/743909891533434718}
\end{center}
\end{document}
