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\begin{document}
\chapter*{AI Airy Ai}
\label{AI}
\section*{AI.1 Introduction}
\label{AI:intro}


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



The initial conditions of AI.1.1 are given at $0$ by
\begin{equation*}
\label{AI:inicond}
\begin{split}
\operatorname{Ai} (0)& =\frac{\sqrt[3]{3}}{3 \Gamma \Bigl(\frac{2}{3}\Bigr)}, \\
\frac{\partial \operatorname{Ai} (x)}{\partial x} (0)& =\frac{-\sqrt[6]{3} \Gamma \Bigl(\frac{2}{3}\Bigr)}{2 \pi}.
\end{split}\tag{AI.1.2}
\end{equation*}



Related function: \href{http://algo.inria.fr/esf/function/BI/BI.html#BI}{Airy Bi}
\section*{AI.2 Series and asymptotic expansions}
\label{AI:asympt}
\subsection*{AI.2.1 Taylor expansion at $0$}
\label{74329225812242019}
\subsubsection*{AI.2.1.1 First terms}
\label{743291359632269852}



\begin{equation*}
\label{AI:asympt:0:RDLBLRDTERMSRDEQ}
\begin{split}
\operatorname{Ai} (x)& =\frac{\sqrt[3]{3}}{3 \Gamma \Bigl(\frac{2}{3}\Bigr)} - \frac{\sqrt[6]{3} \Gamma \Bigl(\frac{2}{3}\Bigr)}{2 \pi}   x + \frac{\sqrt[3]{3}}{18 \Gamma \Bigl(\frac{2}{3}\Bigr)}   x^{3} - \frac{\sqrt[6]{3} \Gamma \Bigl(\frac{2}{3}\Bigr)}{24 \pi}   x^{4} +  \\
& \quad{}\quad{}\frac{\sqrt[3]{3}}{540 \Gamma \Bigl(\frac{2}{3}\Bigr)}   x^{6} - \frac{\sqrt[6]{3} \Gamma \Bigl(\frac{2}{3}\Bigr)}{1008 \pi}   x^{7} + \frac{\sqrt[3]{3}}{38880 \Gamma \Bigl(\frac{2}{3}\Bigr)}   x^{9} - \frac{\sqrt[6]{3} \Gamma \Bigl(\frac{2}{3}\Bigr)}{90720 \pi}    \\
& \quad{}\quad{}x^{10} + \frac{\sqrt[3]{3}}{5132160 \Gamma \Bigl(\frac{2}{3}\Bigr)}   x^{12} - \frac{\sqrt[6]{3} \Gamma \Bigl(\frac{2}{3}\Bigr)}{14152320 \pi}   x^{13} + \frac{\sqrt[3]{3}}{1077753600 \Gamma \Bigl(\frac{2}{3}\Bigr)}    \\
& \quad{}\quad{}x^{15} + \operatorname{O} \bigl(x^{16}\bigr).
\end{split}\tag{AI.2.1.1.1}
\end{equation*}

\subsubsection*{AI.2.1.2 General form}
\label{743292687223946800}



\begin{equation*}
\label{AI:asympt:0:RDLBLRDGENFORMRDEQ}
\begin{split}
\operatorname{Ai} (x)& =\sum_{n = 0}^{\infty} u (n) x^{n}.
\end{split}\tag{AI.2.1.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{AI:asympt:0:toto}
\begin{split}
-u (n) + \bigl(n^{2} + 5 n + 6\bigr) u (n + 3)& =0.
\end{split}\tag{AI.2.1.2.2}
\end{equation*}
Initial conditions of AI.2.1.2.2 are given by
\begin{equation*}
\label{AI:asympt:0:RDLBLRDGENFORMRDIC}
\begin{split}
u (0)& =\frac{\sqrt[3]{3}}{3 \Gamma \Bigl(\frac{2}{3}\Bigr)}, \\
u (1)& =\frac{-\sqrt[6]{3} \Gamma \Bigl(\frac{2}{3}\Bigr)}{2 \pi}, \\
u (2)& =0.
\end{split}\tag{AI.2.1.2.3}
\end{equation*}
The recurrence AI.2.1.2.2 has the closed form solution
\begin{equation*}
\label{AI:asympt:0:RDLBLRDGENFORMRDCLOSED}
\begin{split}
u (3 n)& =\frac{3^{\bigl(\frac{1}{3} - 2 n\bigr)}}{3 \Gamma (n + 1) \Gamma \Bigl(n + \frac{2}{3}\Bigr)}, \\
u (3 n + 1)& =\frac{-3^{\bigl(\frac{2}{3} - 2 n\bigr)}}{9 \Gamma (n + 1) \Gamma \Bigl(n + \frac{4}{3}\Bigr)}, \\
u (3 n + 2)& =0.
\end{split}\tag{AI.2.1.2.4}
\end{equation*}

\subsection*{AI.2.2 Asymptotic expansion at $\infty$}
\label{74329297396414947}
\subsubsection*{AI.2.2.1 First terms}
\label{743292121022573901}

\begin{equation*}
\begin{split}
& \operatorname{Ai} (x)\approx \operatorname{e} ^{\biggl(\frac{-2}{3 \xi^{3}}\biggr)} \sqrt{\xi} \Biggl(\frac{1}{2 \sqrt{\pi}} - \frac{5 \xi^{3}}{96 \sqrt{\pi}} + \ldots\Biggr)
\end{split}
\end{equation*}
where $\xi = \sqrt{\frac{1}{x}}$\subsubsection*{AI.2.2.2 General form}
\label{743292665288968349}

\begin{equation*}
\begin{split}
& \operatorname{Ai} (x)\approx \operatorname{e} ^{\biggl(\frac{-2}{3 \xi^{3}}\biggr)} \sqrt{\xi} \sum_{n = 0}^{\infty} u (n) \xi^{n}
\end{split}
\end{equation*}
where $\xi = \sqrt{\frac{1}{x}}$The coefficients $u (n)$ satisfy the following recurrence
\begin{equation*}
\begin{split}
16 u (n) n + u (n - 3) \bigl(-31 + 12 n + 4 (n - 3)^{2}\bigr)& =0.
\end{split}
\end{equation*}
whose initial conditions are given by
\begin{equation*}
\begin{split}
u (0)& =\frac{1}{2 \sqrt{\pi}}, \\
u (1)& =0, \\
u (2)& =0.
\end{split}
\end{equation*}
This recurrence has the closed form solution
\begin{equation*}
\begin{split}
u (3 n)& =\frac{(-1)^{n} 6^{(2 n)} \Gamma \Bigl(n + \frac{5}{6}\Bigr) \Gamma \Bigl(n + \frac{1}{6}\Bigr)}{4 \pi^{\frac{3}{2}} 48^{n} \Gamma (n + 1)}, \\
u (3 n + 1)& =0, \\
u (3 n + 2)& =0.
\end{split}
\end{equation*}
\section*{AI.3 Graphs}
\label{743303134050365811}
\subsection*{AI.3.1 Real axis}
\label{743303987785640265}
\begin{center}
\includegraphics[width=6cm]{AI/743837372151835203}
\end{center}
\subsection*{AI.3.2 Complex plane}
\label{743303674198272844}
\begin{center}
\includegraphics[width=6cm]{AI/743837102821615215}
\end{center}
\end{document}