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


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



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



Related functions: \href{http://algo.inria.fr/esf/function/ASN/ASN.html#ASN}{Inverse Sine},\href{http://algo.inria.fr/esf/function/HACS/HACS.html#HACS}{Inverse Hyperbolic Cosine}
\section*{ACS.2 Series and asymptotic expansions}
\label{ACS:asympt}
\subsection*{ACS.2.1 Asymptotic expansion at $-1$}
\label{743278131905754565}
\subsubsection*{ACS.2.1.1 First terms}
\label{ACS:asympt:TB1:termsec}



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

\subsubsection*{ACS.2.1.2 General form}
\label{743278643842443844}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{ACS.2.2 Asymptotic expansion at $1$}
\label{74327874813656220}
\subsubsection*{ACS.2.2.1 First terms}
\label{ACS:asympt:1:termsec}



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

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



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

\subsection*{ACS.2.3 Asymptotic expansion at $\infty$}
\label{743278135537108795}
\subsubsection*{ACS.2.3.1 First terms}
\label{ACS:asympt:infinity:termsec}



\begin{equation*}
\label{ACS:asympt:infinity:terms}
\begin{split}
& \operatorname{arccos} (x)\approx  \\
& \quad{}\quad{}\Biggl(i \operatorname{ln} (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} \biggl(\frac{1}{x}\biggr)\ldots\Biggr).
\end{split}\tag{ACS.2.3.1.1}
\end{equation*}

\subsubsection*{ACS.2.3.2 General form}
\label{743278672075358391}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{ACS.2.4 Taylor expansion at $0$}
\label{743277155590763466}
\subsubsection*{ACS.2.4.1 First terms}
\label{74327788430571674}



\begin{equation*}
\label{ACS:asympt:0:RDLBLRDTERMSRDEQ}
\begin{split}
\operatorname{arccos} (x)& =\frac{\pi}{2} - x - \frac{1}{6}   x^{3} - \frac{3}{40}   x^{5} - \frac{5}{112}   x^{7} - \frac{35}{1152}   x^{9} - \frac{63}{2816}   x^{11} - \frac{231}{13312}    \\
& \quad{}\quad{}x^{13} - \frac{143}{10240}   x^{15} + \operatorname{O} \bigl(x^{16}\bigr).
\end{split}\tag{ACS.2.4.1.1}
\end{equation*}

\subsubsection*{ACS.2.4.2 General form}
\label{743277146486307198}



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

\section*{ACS.3 Graphs}
\label{743283742867231360}
\subsection*{ACS.3.1 Real axis}
\label{743283991638155474}
\begin{center}
\includegraphics[width=6cm]{ACS/743743324818617759}
\end{center}
\subsection*{ACS.3.2 Complex plane}
\label{743283452610874309}
\begin{center}
\includegraphics[width=6cm]{ACS/743743426782336744}
\end{center}
\end{document}