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\begin{document}
\chapter*{CI Cosine Integral}
\label{CI}
\section*{CI.1 Introduction}
\label{CI:intro}


Let $x$ be a complex variable of $\mathbb{C} \setminus \{0,\infty\}$.The function Cosine Integral (noted $\operatorname{Ci}$) is defined by the following third order differential equation
\begin{equation*}
\label{CI:diffeq}
\begin{split}
x \frac{\partial y (x)}{\partial x} + 2 \frac{\partial^{2} y (x)}{\partial x^{2}} + x \frac{\partial^{3} y (x)}{\partial x^{3}}& =0.
\end{split}\tag{CI.1.1}
\end{equation*}



The initial conditions of CI.1.1 at $0$ are not simple to state, since $0$ is a (regular) singular point.


Related function: \href{http://algo.inria.fr/esf/function/SI/SI.html#SI}{Sine Integral}
\section*{CI.2 Series and asymptotic expansions}
\label{CI:asympt}
\subsection*{CI.2.1 Asymptotic expansion at $0$}
\label{743345628962106460}
\subsubsection*{CI.2.1.1 First terms}
\label{CI:asympt:0:termsec}



\begin{equation*}
\label{CI:asympt:0:terms}
\begin{split}
& \operatorname{Ci} (x)\approx \Biggl(\frac{x^{2}}{4} - \frac{x^{4}}{96} + \frac{x^{6}}{4320} - \frac{x^{8}}{322560} - \operatorname{ln} (x) + \gamma\ldots\Biggr).
\end{split}\tag{CI.2.1.1.1}
\end{equation*}

\subsubsection*{CI.2.1.2 General form}
\label{743345830196465066}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{CI.2.2 Asymptotic expansion at $\infty$}
\label{743347854564102988}
\subsubsection*{CI.2.2.1 First terms}
\label{743347579030892770}



\begin{equation*}
\begin{split}
& \operatorname{Ci} (x)\approx \operatorname{e} ^{\biggl(-\frac{\operatorname{RootOf} _{\xi,1} (1 + \xi^{2})}{x}\biggr)} x y _{0} (x) +  \\
& \quad{}\quad{}\operatorname{e} ^{\biggl(-\frac{\operatorname{RootOf} _{\xi,2} (1 + \xi^{2})}{x}\biggr)} x y _{1} (x),
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
y _{0} (x)& =\frac{i}{2} + \frac{i}{2} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) x +  \\
& \quad{}\quad{}\biggl(\frac{i}{4} + \frac{5 i}{4} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}\biggr) x^{2} + \Biggl(i \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) +  \\
& \quad{}\quad{}4 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \biggl(\frac{i}{4} + \frac{5 i}{4} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}\biggr)\Biggr) x^{3} + 2 \ldots \\
y _{1} (x)& =\frac{-i}{2} - \frac{i}{2} \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) x -  \\
& \quad{}\quad{}-\biggl(\frac{-i}{4} - \frac{5 i}{4} \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}\biggr) x^{2} - -\Biggl(-i\operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) +  \\
& \quad{}\quad{}4 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \biggl(\frac{-i}{4} - \frac{5 i}{4} \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{2}\biggr)\Biggr) x^{3} +  \\
& \quad{}\quad{}2 \ldots
\end{split}
\end{equation*}

\subsubsection*{CI.2.2.2 General form}
\label{743347354525039229}
\paragraph*{CI.2.2.2.1 Auxiliary function $y _{0} (x)$}
\label{743347689487903655}
The coefficients $u (n)$ of $y _{0} (x)$ satisfy the following recurrence
\begin{equation*}
\begin{split}
& 2 u (n) n + u (n - 1) \Bigl(-2\operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) -  \\
& 5 \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) (n - 1) - 3 (n - 1)^{2} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)\Bigr) +  \\
& u (n - 2) \bigl(8 - 5 n - 4 (n - 2)^{2} - (n - 2)^{3}\bigr)=0
\end{split}
\end{equation*}
whose initial conditions are given by
\begin{equation*}
\begin{split}
u (1)& =\frac{i}{2} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) \\
u (0)& =\frac{i}{2}
\end{split}
\end{equation*}
\paragraph*{CI.2.2.2.2 Auxiliary function $y _{1} (x)$}
\label{743347474889986267}
The coefficients $u (n)$ of $y _{1} (x)$ satisfy the following recurrence
\begin{equation*}
\begin{split}
& 2 u (n) n + u (n - 1) \Bigl(-2\operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) -  \\
& 5 \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) (n - 1) - 3 (n - 1)^{2} \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)\Bigr) +  \\
& u (n - 2) \bigl(8 - 5 n - 4 (n - 2)^{2} - (n - 2)^{3}\bigr)=0
\end{split}
\end{equation*}
whose initial conditions are given by
\begin{equation*}
\begin{split}
u (1)& =-\frac{i}{2}\operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) \\
u (0)& =\frac{-i}{2}
\end{split}
\end{equation*}
\end{document}