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\begin{document}
\chapter*{ACT Inverse Cotangent}
\label{ACT}
\section*{ACT.1 Introduction}
\label{ACT:intro}


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



The initial conditions of ACT.1.1 are given at $0$ by
\begin{equation*}
\label{ACT:inicond}
\begin{split}
\operatorname{arccot} (0)& =\frac{\pi}{2}, \\
\frac{\partial \operatorname{arccot} (x)}{\partial x} (0)& =-1.
\end{split}\tag{ACT.1.2}
\end{equation*}

\section*{ACT.2 Series and asymptotic expansions}
\label{ACT:asympt}
\subsection*{ACT.2.1 Asymptotic expansion at $-i$}
\label{74328480703111299}
\subsubsection*{ACT.2.1.1 First terms}
\label{ACT:asympt:TBI:termsec}



\begin{equation*}
\label{ACT:asympt:TBI:terms}
\begin{split}
& \operatorname{arccot} (x)\approx \Biggl(\frac{\pi}{2} + \frac{i}{2} \operatorname{ln} (2) +  \\
& \quad{}\quad{}\frac{i}{4} \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \Bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr) -  \\
& \quad{}\quad{}\frac{i}{16} \Bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr)^{2} + \frac{i \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{i \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{i \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{i \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{i \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{i \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{i}{2} \operatorname{ln} \Bigl(x - \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr)\ldots\Biggr).
\end{split}\tag{ACT.2.1.1.1}
\end{equation*}

\subsubsection*{ACT.2.1.2 General form}
\label{743284769034410057}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{ACT.2.2 Asymptotic expansion at $i$}
\label{743284552921209928}
\subsubsection*{ACT.2.2.1 First terms}
\label{ACT:asympt:I:termsec}



\begin{equation*}
\label{ACT:asympt:I:terms}
\begin{split}
& \operatorname{arccot} (x)\approx \Biggl(\frac{\pi}{2} - \frac{i}{2} \operatorname{ln} (2) -  \\
& \quad{}\quad{}\frac{i}{4} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \Bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr) +  \\
& \quad{}\quad{}\frac{i}{16} \Bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr)^{2} - \frac{i \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{i \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{i \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{i \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{i \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{i \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{i}{2} \operatorname{ln} \Bigl(x - \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr)\ldots\Biggr).
\end{split}\tag{ACT.2.2.1.1}
\end{equation*}

\subsubsection*{ACT.2.2.2 General form}
\label{743284605127114594}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{ACT.2.3 Taylor expansion at $0$}
\label{743284454100745237}
\subsubsection*{ACT.2.3.1 First terms}
\label{743284340807728335}



\begin{equation*}
\label{ACT:asympt:0:RDLBLRDTERMSRDEQ}
\begin{split}
\operatorname{arccot} (x)& =\frac{\pi}{2} - 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{ACT.2.3.1.1}
\end{equation*}

\subsubsection*{ACT.2.3.2 General form}
\label{743284281314862289}



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

\section*{ACT.3 Graphs}
\label{743288912854227507}
\subsection*{ACT.3.1 Real axis}
\label{743288891515242515}
\begin{center}
\includegraphics[width=6cm]{ACT/743768259838341826}
\end{center}
\subsection*{ACT.3.2 Complex plane}
\label{743288620324828055}
\begin{center}
\includegraphics[width=6cm]{ACT/74376846341480775}
\end{center}
\end{document}