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\begin{document}
\chapter*{BSI Bessel I}
\label{BSI}
\section*{BSI.1 Introduction}
\label{BSI:intro}


Let $x$ be a complex variable of $\mathbb{C} \setminus \{0,\infty\}$ and let $\nu$ denote a parameter (independent of $x$).The function Bessel I (noted $\operatorname{I} _{\nu}$) is defined by the following second order differential equation
\begin{equation*}
\label{BSI:diffeq}
\begin{split}
-x^{2} + \nu^{2}y (x) + x \frac{\partial y (x)}{\partial x} + x^{2} \frac{\partial^{2} y (x)}{\partial x^{2}}& =0.
\end{split}\tag{BSI.1.1}
\end{equation*}



Although $0$ is a singularity of BSI.1.1, the initial conditions can be given by 
\begin{equation*}
\label{BSI:inicond}
\begin{split}
\frac{\partial \frac{\operatorname{I} _{\nu} (x)}{x^{\nu}}}{\partial x}& =\frac{1}{\Gamma (\nu + 1) 2^{\nu}}.
\end{split}\tag{BSI.1.2}
\end{equation*}



The formulae of this document are valid for $-2\nu \not\in \mathbb{Z} .$


Related function: \href{http://algo.inria.fr/esf/function/BSK/BSK.html#BSK}{Bessel K}
\section*{BSI.2 Series and asymptotic expansions}
\label{BSI:asympt}
\subsection*{BSI.2.1 Asymptotic expansion at $\infty$}
\label{743334267866007697}
\subsubsection*{BSI.2.1.1 First terms}
\label{743334681919512091}



\begin{equation*}
\begin{split}
& \operatorname{I} _{\nu} (x)\approx \operatorname{e} ^{\Bigl(-\frac{1}{x}\Bigr)} \sqrt{x} y _{0} (x),
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
y _{0} (x)& =\frac{\sqrt{2}}{2 \sqrt{\pi}} - \frac{-\bigl(4 \nu^{2} - 1\bigr) \sqrt{2} x}{16 \sqrt{\pi}} + \frac{\bigl(4 \nu^{2} - 9\bigr) \bigl(4 \nu^{2} - 1\bigr) \sqrt{2} x^{2}}{256 \sqrt{\pi}} -  \\
& \quad{}\quad{}\frac{-\bigl(4 \nu^{2} - 25\bigr) \bigl(4 \nu^{2} - 9\bigr) \bigl(4 \nu^{2} - 1\bigr) \sqrt{2} x^{3}}{6144 \sqrt{\pi}} + 2 \ldots
\end{split}
\end{equation*}

\subsubsection*{BSI.2.1.2 General form}
\label{743334197660018541}
\paragraph*{BSI.2.1.2.1 Auxiliary function $y _{0} (x)$}
\label{743334366516647943}
The coefficients $u (n)$ of $y _{0} (x)$ satisfy the following recurrence
\begin{equation*}
\begin{split}
8 u (n) n + u (n - 1) \bigl(-4\nu^{2} - 3 + 4 n + 4 (n - 1)^{2}\bigr)& =0
\end{split}
\end{equation*}
whose initial conditions are given by
\begin{equation*}
\begin{split}
u (0)& =\frac{\sqrt{2}}{2 \sqrt{\pi}}
\end{split}
\end{equation*}
This recurrence has the closed form solution
\begin{equation*}
\begin{split}
u (n)& =\frac{2^{\bigl(n + \frac{1}{2}\bigr)} \Gamma \Bigl(n + \nu + \frac{1}{2}\Bigr) \Gamma \Bigl(n - \nu + \frac{1}{2}\Bigr) \operatorname{sin} \biggl(\frac{\pi (2 \nu + 1)}{2}\biggr) (-2)^{n}}{2 8^{n} \pi^{\frac{3}{2}} \Gamma (n + 1)}.
\end{split}
\end{equation*}
\subsection*{BSI.2.2 Asymptotic expansion at $0$}
\label{743333785598722235}
\subsubsection*{BSI.2.2.1 First terms}
\label{BSI:asympt:0:termsec}



\begin{equation*}
\label{BSI:asympt:0:terms}
\begin{split}
& \operatorname{I} _{\nu} (x)\approx x^{\nu} \Biggl(\frac{1}{\Gamma (\nu + 1) 2^{\nu}} + \frac{x^{2}}{4 \Gamma (\nu + 1) 2^{\nu} (\nu + 1)} +  \\
& \quad{}\quad{}\frac{x^{4}}{32 \Gamma (\nu + 1) 2^{\nu} (\nu + 1) (\nu + 2)} +  \\
& \quad{}\quad{}\frac{x^{6}}{384 \Gamma (\nu + 1) 2^{\nu} (\nu + 1) (\nu + 2) (\nu + 3)}\ldots\Biggr).
\end{split}\tag{BSI.2.2.1.1}
\end{equation*}

\subsubsection*{BSI.2.2.2 General form}
\label{BSI:asympt:0:genf}



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

\end{document}