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\begin{document}
\chapter*{LI2 Dilogarithm}
\label{LI2}
\section*{LI2.1 Introduction}
\label{LI2:intro}


Let $x$ be a complex variable of $\mathbb{C} \setminus \{0,\infty\}$.The function Dilogarithm (noted $\operatorname{dilog}$) is defined by the following third order differential equation
\begin{equation*}
\label{LI2:diffeq}
\begin{split}
\frac{\partial y (x)}{\partial x} - -(-1 + 3 x) \frac{\partial^{2} y (x)}{\partial x^{2}} - -\bigl(-x + x^{2}\bigr) \frac{\partial^{3} y (x)}{\partial x^{3}}& =0.
\end{split}\tag{LI2.1.1}
\end{equation*}



The initial conditions of LI2.1.1 at $0$ are not simple to state, since $0$ is a (regular) singular point.
\section*{LI2.2 Series and asymptotic expansions}
\label{LI2:asympt}
\subsection*{LI2.2.1 Asymptotic expansion at $1$}
\label{743638579791479112}
\subsubsection*{LI2.2.1.1 First terms}
\label{LI2:asympt:1:termsec}



\begin{equation*}
\label{LI2:asympt:1:terms}
\begin{split}
& \operatorname{dilog} (x)\approx (x - 1) \Biggl(-\frac{5}{4} + \frac{x}{4} - \frac{(x - 1)^{2}}{9} + \frac{(x - 1)^{3}}{16} - \frac{(x - 1)^{4}}{25} + \frac{(x - 1)^{5}}{36} -  \\
& \quad{}\quad{}\frac{(x - 1)^{6}}{49} + \frac{(x - 1)^{7}}{64} - \frac{(x - 1)^{8}}{81} + \frac{(x - 1)^{9}}{100} - \frac{(x - 1)^{10}}{121} + \frac{(x - 1)^{11}}{144} -  \\
& \quad{}\quad{}\frac{(x - 1)^{12}}{169} + \frac{(x - 1)^{13}}{196} - \frac{(x - 1)^{14}}{225} + \frac{(x - 1)^{15}}{256}\ldots\Biggr).
\end{split}\tag{LI2.2.1.1.1}
\end{equation*}

\subsubsection*{LI2.2.1.2 General form}
\label{LI2:asympt:1:genf}



\begin{equation*}
\label{LI2:asympt:1:genfsum}
\begin{split}
& \operatorname{dilog} (x)\approx (x - 1) \sum_{n = 0}^{\infty} u (n) (x - 1)^{n}.
\end{split}\tag{LI2.2.1.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{LI2:asympt:1:genfrec}
\begin{split}
u (n) (n + 1)^{2} n + u (n - 1) n^{3}& =0.
\end{split}\tag{LI2.2.1.2.2}
\end{equation*}
Initial conditions of LI2.2.1.2.2 are given by
\begin{equation*}
\label{LI2:asympt:1:genfic}
\begin{split}
u (0)& =-1, \\
u (1)& =\frac{1}{4}.
\end{split}\tag{LI2.2.1.2.3}
\end{equation*}
The recurrence LI2.2.1.2.2 has the closed form solution
\begin{equation*}
\label{LI2:asympt:1:RDINREFRDGENFROMRDCLOSED}
\begin{split}
u (n)& =-\frac{(-1)^{n}}{(n + 1)^{2}}.
\end{split}\tag{LI2.2.1.2.4}
\end{equation*}

\subsection*{LI2.2.2 Asymptotic expansion at $\infty$}
\label{743638740993658161}
\subsubsection*{LI2.2.2.1 First terms}
\label{LI2:asympt:infinity:termsec}



\begin{equation*}
\label{LI2:asympt:infinity:terms}
\begin{split}
& \operatorname{dilog} (x)\approx \Biggl(\frac{1 - 8 \operatorname{ln} \Bigl(\frac{1}{x}\Bigr)}{64 x^{8}} + \frac{1 - 7 \operatorname{ln} \Bigl(\frac{1}{x}\Bigr)}{49 x^{7}} + \frac{1 - 6 \operatorname{ln} \Bigl(\frac{1}{x}\Bigr)}{36 x^{6}} + \frac{1 - 5 \operatorname{ln} \Bigl(\frac{1}{x}\Bigr)}{25 x^{5}} -  \\
& \quad{}\quad{}-\frac{-4\operatorname{ln} \Bigl(\frac{1}{x}\Bigr) + 1}{16 x^{4}} + \frac{1 - 3 \operatorname{ln} \Bigl(\frac{1}{x}\Bigr)}{9 x^{3}} + \frac{1 - 2 \operatorname{ln} \Bigl(\frac{1}{x}\Bigr)}{4 x^{2}} + \frac{1 - \operatorname{ln} \Bigl(\frac{1}{x}\Bigr)}{x} - \frac{\pi^{2}}{6} -  \\
& \quad{}\quad{}\frac{\operatorname{ln} \Bigl(\frac{1}{x}\Bigr)^{2}}{2}\ldots\Biggr).
\end{split}\tag{LI2.2.2.1.1}
\end{equation*}

\subsubsection*{LI2.2.2.2 General form}
\label{743638760520839612}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\subsection*{LI2.2.3 Asymptotic expansion at $0$}
\label{743637445179899520}
\subsubsection*{LI2.2.3.1 First terms}
\label{LI2:asympt:0:termsec}



\begin{equation*}
\label{LI2:asympt:0:terms}
\begin{split}
& \operatorname{dilog} (x)\approx \Biggl(-\frac{15}{64} + \frac{\operatorname{ln} (x)}{8}x^{8} - -\Biggl(-\frac{13}{49} - \frac{\operatorname{ln} (x)}{7}\Biggr) x^{7} -  \\
& \quad{}\quad{}-\Biggl(-\frac{11}{36} - \frac{\operatorname{ln} (x)}{6}\Biggr) x^{6} - -\Biggl(-\frac{9}{25} - \frac{\operatorname{ln} (x)}{5}\Biggr) x^{5} - -\Biggl(-\frac{7}{16} - \frac{\operatorname{ln} (x)}{4}\Biggr) x^{4} -  \\
& \quad{}\quad{}-\Biggl(-\frac{5}{9} - \frac{\operatorname{ln} (x)}{3}\Biggr) x^{3} - -\Biggl(-\frac{3}{4} - \frac{\operatorname{ln} (x)}{2}\Biggr) x^{2} - -\bigl(-1 - \operatorname{ln} (x)\bigr) x + \frac{\pi^{2}}{6}\ldots\Biggr).
\end{split}\tag{LI2.2.3.1.1}
\end{equation*}

\subsubsection*{LI2.2.3.2 General form}
\label{743637601993064528}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\end{document}