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\begin{document}
\chapter*{DBI Derivative of Airy Bi}
\label{DBI}
\section*{DBI.1 Introduction}
\label{DBI:intro}


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



Although $0$ is a singularity of DBI.1.1, the initial conditions can be given by 
\begin{equation*}
\label{DBI:inicond}
\begin{split}
[1] \operatorname{Bi} \prime (x)& =\frac{3^{\frac{2}{3}} \Gamma \Bigl(\frac{2}{3}\Bigr)}{2 \pi}, \\
\bigl[x^{2}\bigr] \operatorname{Bi} \prime (x)& =\frac{3^{\frac{5}{6}}}{6 \Gamma \Bigl(\frac{2}{3}\Bigr)}.
\end{split}\tag{DBI.1.2}
\end{equation*}



Related function: \href{http://algo.inria.fr/esf/function/DAI/DAI.html#DAI}{Derivative of Airy Ai}
\section*{DBI.2 Series and asymptotic expansions}
\label{DBI:asympt}
\subsection*{DBI.2.1 Asymptotic expansion at $\infty$}
\label{74335734452291350}
\subsubsection*{DBI.2.1.1 First terms}
\label{743357600170500569}

\begin{equation*}
\begin{split}
& \operatorname{Bi} \prime (x)\approx \frac{\operatorname{e} ^{\Bigl(\frac{-2}{3 \xi^{3}}\Bigr)} \biggl(\frac{i}{\sqrt{\pi}} + \frac{7 i \xi^{3}}{48 \sqrt{\pi}} + \ldots\biggr)}{\sqrt{\xi}}
\end{split}
\end{equation*}
where $\xi = -\sqrt{\frac{1}{x}}$\subsubsection*{DBI.2.1.2 General form}
\label{743357201873651317}

\begin{equation*}
\begin{split}
& \operatorname{Bi} \prime (x)\approx \frac{\operatorname{e} ^{\Bigl(\frac{-2}{3 \xi^{3}}\Bigr)} \sum_{n = 0}^{\infty} u (n) \xi^{n}}{\sqrt{\xi}}
\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(-43 + 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{i}{\sqrt{\pi}}, \\
u (2)& =0, \\
u (1)& =0.
\end{split}
\end{equation*}
This recurrence has the closed form solution
\begin{equation*}
\begin{split}
u (3 n + 1)& =0, \\
u (3 n + 2)& =0, \\
u (3 n)& =\frac{-i(-1)^{n} 6^{(2 n)} \Gamma \Bigl(n + \frac{7}{6}\Bigr) \Gamma \Bigl(n - \frac{1}{6}\Bigr)}{2 \pi^{\frac{3}{2}} 48^{n} \Gamma (n + 1)}.
\end{split}
\end{equation*}
\subsection*{DBI.2.2 Asymptotic expansion at $0$}
\label{743357552176312846}
\subsubsection*{DBI.2.2.1 First terms}
\label{DBI:asympt:0:termsec}



\begin{equation*}
\label{DBI:asympt:0:terms}
\begin{split}
& \operatorname{Bi} \prime (x)\approx \Biggl(\frac{3^{\frac{5}{6}} x^{8}}{4320 \Gamma \Bigl(\frac{2}{3}\Bigr)} + \frac{x^{6} 3^{\frac{2}{3}} \Gamma \Bigl(\frac{2}{3}\Bigr)}{144 \pi} + \frac{3^{\frac{5}{6}} x^{5}}{90 \Gamma \Bigl(\frac{2}{3}\Bigr)} + \frac{x^{3} 3^{\frac{2}{3}} \Gamma \Bigl(\frac{2}{3}\Bigr)}{6 \pi} +  \\
& \quad{}\quad{}\frac{3^{\frac{5}{6}} x^{2}}{6 \Gamma \Bigl(\frac{2}{3}\Bigr)} + \frac{3^{\frac{2}{3}} \Gamma \Bigl(\frac{2}{3}\Bigr)}{2 \pi}\ldots\Biggr).
\end{split}\tag{DBI.2.2.1.1}
\end{equation*}

\subsubsection*{DBI.2.2.2 General form}
\label{743357290669948665}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\end{document}
