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\begin{document}
\chapter*{BSY Bessel Y}
\label{BSY}
\section*{BSY.1 Introduction}
\label{BSY: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 Y (noted $\operatorname{Y} _{\nu}$) is defined by the following second order differential equation
\begin{equation*}
\label{BSY:diffeq}
\begin{split}
\bigl(x^{2} - \nu^{2}\bigr) y (x) + x \frac{\partial y (x)}{\partial x} + x^{2} \frac{\partial^{2} y (x)}{\partial x^{2}}& =0.
\end{split}\tag{BSY.1.1}
\end{equation*}



Although $0$ is a singularity of BSY.1.1, the initial conditions can be given by 
\begin{equation*}
\label{BSY:inicond}
\begin{split}
\Bigl[x^{(-\nu)}\Bigr] \operatorname{Y} _{\nu} (x)& =-\frac{1}{\frac{\Gamma (-\nu + 1) \operatorname{sin} (\nu \pi)}{2^{\nu}}}, \\
\bigl[x^{\nu}\bigr] \operatorname{Y} _{\nu} (x)& =\frac{\operatorname{cos} (\nu \pi)}{\Gamma (\nu + 1) 2^{\nu} \operatorname{sin} (\nu \pi)}.
\end{split}\tag{BSY.1.2}
\end{equation*}



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


Related functions: \href{http://algo.inria.fr/esf/function/H1/H1.html#H1}{Hankel H1},\href{http://algo.inria.fr/esf/function/H2/H2.html#H2}{Hankel H2},\href{http://algo.inria.fr/esf/function/BSJ/BSJ.html#BSJ}{Bessel J}
\section*{BSY.2 Series and asymptotic expansions}
\label{BSY:asympt}
\subsection*{BSY.2.1 Asymptotic expansion at $\infty$}
\label{743340670169458329}
\subsubsection*{BSY.2.1.1 First terms}
\label{743340323454853601}



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

\subsubsection*{BSY.2.1.2 General form}
\label{743340737380068865}
\paragraph*{BSY.2.1.2.1 Auxiliary function $y _{0} (x)$}
\label{743340571078647666}
The coefficients $u (n)$ of $y _{0} (x)$ satisfy the following recurrence
\begin{equation*}
\begin{split}
& 8 u (n) n \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) + 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{-i\sqrt{2} \operatorname{e} ^{\bigl(-\frac{i}{4}\pi (2 \nu + 1)\bigr)}}{2 \sqrt{\pi}}
\end{split}
\end{equation*}
This recurrence has the closed form solution
\begin{equation*}
\begin{split}
u (n)& =\Biggl(-i2^{\Bigl(n + \frac{1}{2}\Bigr)} \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{n} \Gamma \biggl(\nu + \frac{1}{2} + n\biggr) \Gamma \biggl(n - \nu + \frac{1}{2}\biggr)  \\
& \quad{}\quad{}\operatorname{sin} \Biggl(\frac{\pi (2 \nu + 1)}{2}\Biggr) (-2)^{n} (-1)^{n} \operatorname{e} ^{\Bigl(-\frac{i}{4}\pi (2 \nu + 1)\Bigr)}\Biggr)\Bigg/ \\
& \quad{}\quad{}\Biggl(2 8^{n} \pi^{\frac{3}{2}} \Gamma (n + 1)\Biggr).
\end{split}
\end{equation*}
\paragraph*{BSY.2.1.2.2 Auxiliary function $y _{1} (x)$}
\label{743340608974660159}
The coefficients $u (n)$ of $y _{1} (x)$ satisfy the following recurrence
\begin{equation*}
\begin{split}
& 8 u (n) n \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr) + 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{i \sqrt{2} \operatorname{e} ^{\frac{i}{4} \pi (2 \nu + 1)}}{2 \sqrt{\pi}}
\end{split}
\end{equation*}
This recurrence has the closed form solution
\begin{equation*}
\begin{split}
u (n)& =\Biggl(i 2^{\Bigl(n + \frac{1}{2}\Bigr)} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{n} \Gamma \biggl(\nu + \frac{1}{2} + n\biggr) \Gamma \biggl(n - \nu + \frac{1}{2}\biggr)  \\
& \quad{}\quad{}\operatorname{sin} \Biggl(\frac{\pi (2 \nu + 1)}{2}\Biggr) (-2)^{n} (-1)^{n} \operatorname{e} ^{\frac{i}{4} \pi (2 \nu + 1)}\Biggr)\Bigg/ \\
& \quad{}\quad{}\Biggl(2 8^{n} \pi^{\frac{3}{2}} \Gamma (n + 1)\Biggr).
\end{split}
\end{equation*}
\subsection*{BSY.2.2 Asymptotic expansion at $0$}
\label{743339248634826183}
\subsubsection*{BSY.2.2.1 First terms}
\label{BSY:asympt:0:termsec}



\begin{equation*}
\label{BSY:asympt:0:terms}
\begin{split}
& \operatorname{Y} _{\nu} (x)\approx \Biggl(-\frac{1}{\frac{\Gamma (-\nu + 1) \operatorname{sin} (\nu \pi)}{2^{\nu}}} - \frac{x^{2}}{\frac{4 (\nu - 1) \Gamma (-\nu + 1) \operatorname{sin} (\nu \pi)}{2^{\nu}}} -  \\
& \quad{}\quad{}\frac{x^{4}}{\frac{32 (\nu - 1) (\nu - 2) \Gamma (-\nu + 1) \operatorname{sin} (\nu \pi)}{2^{\nu}}} -  \\
& \quad{}\quad{}\frac{x^{6}}{\frac{384 (\nu - 1) (\nu - 2) (\nu - 3) \Gamma (-\nu + 1) \operatorname{sin} (\nu \pi)}{2^{\nu}}} -  \\
& \quad{}\quad{}\frac{x^{8}}{\frac{6144 (\nu - 1) (\nu - 2) (\nu - 3) (\nu - 4) \Gamma (-\nu + 1) \operatorname{sin} (\nu \pi)}{2^{\nu}}}\ldots\Biggr)\Bigg/x^{\nu} +  \\
& \quad{}\quad{}x^{\nu} \Biggl(\frac{\operatorname{cos} (\nu \pi)}{\Gamma (\nu + 1) 2^{\nu} \operatorname{sin} (\nu \pi)} - \frac{x^{2} \operatorname{cos} (\nu \pi)}{4 (\nu + 1) \Gamma (\nu + 1) 2^{\nu} \operatorname{sin} (\nu \pi)}  \\
& \quad{}\quad{}+ \frac{x^{4} \operatorname{cos} (\nu \pi)}{32 (\nu + 1) (\nu + 2) \Gamma (\nu + 1) 2^{\nu} \operatorname{sin} (\nu \pi)} -  \\
& \quad{}\quad{}\frac{x^{6} \operatorname{cos} (\nu \pi)}{384 (\nu + 1) (\nu + 2) (\nu + 3) \Gamma (\nu + 1) 2^{\nu} \operatorname{sin} (\nu \pi)} +  \\
& \quad{}\quad{}\frac{x^{8} \operatorname{cos} (\nu \pi)}{6144 (\nu + 1) (\nu + 2) (\nu + 3) (\nu + 4) \Gamma (\nu + 1) 2^{\nu} \operatorname{sin} (\nu \pi)}\ldots\Biggr).
\end{split}\tag{BSY.2.2.1.1}
\end{equation*}

\subsubsection*{BSY.2.2.2 General form}
\label{743339142684554931}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\end{document}