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\begin{document}
\chapter*{BSK Bessel K}
\label{BSK}
\section*{BSK.1 Introduction}
\label{BSK: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 K (noted $\operatorname{K} _{\nu}$) is defined by the following second order differential equation
\begin{equation*}
\label{BSK: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{BSK.1.1}
\end{equation*}



Although $0$ is a singularity of BSK.1.1, the initial conditions can be given by 
\begin{equation*}
\label{BSK:inicond}
\begin{split}
\Bigl[x^{(-\nu)}\Bigr] \operatorname{K} _{\nu} (x)& =2^{(\nu - 1)} \Gamma (\nu), \\
\bigl[x^{\nu}\bigr] \operatorname{K} _{\nu} (x)& =-\frac{\pi}{\Gamma (\mu + 1) \operatorname{sin} (\mu \pi) 2^{(\mu + 1)}}.
\end{split}\tag{BSK.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/BSI/BSI.html#BSI}{Bessel I}
\section*{BSK.2 Series and asymptotic expansions}
\label{BSK:asympt}
\subsection*{BSK.2.1 Asymptotic expansion at $\infty$}
\label{743338845112505300}
\subsubsection*{BSK.2.1.1 First terms}
\label{743338384393855897}



\begin{equation*}
\begin{split}
& \operatorname{K} _{\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} \sqrt{\pi}}{2} - \frac{-\bigl(4 \nu^{2} - 1\bigr) \sqrt{2} \sqrt{\pi} x}{16} +  \\
& \quad{}\quad{}\frac{\bigl(4 \nu^{2} - 9\bigr) \bigl(4 \nu^{2} - 1\bigr) \sqrt{2} \sqrt{\pi} x^{2}}{256} -  \\
& \quad{}\quad{}\frac{-\bigl(4 \nu^{2} - 25\bigr) \bigl(4 \nu^{2} - 9\bigr) \bigl(4 \nu^{2} - 1\bigr) \sqrt{2} \sqrt{\pi} x^{3}}{6144} + 2 \ldots
\end{split}
\end{equation*}

\subsubsection*{BSK.2.1.2 General form}
\label{74333828078765810}
\paragraph*{BSK.2.1.2.1 Auxiliary function $y _{0} (x)$}
\label{743338580793905552}
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} \sqrt{\pi}}{2}
\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} \sqrt{\pi} \Gamma (n + 1)}.
\end{split}
\end{equation*}
\subsection*{BSK.2.2 Asymptotic expansion at $0$}
\label{743338786809332918}
\subsubsection*{BSK.2.2.1 First terms}
\label{BSK:asympt:0:termsec}



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

\subsubsection*{BSK.2.2.2 General form}
\label{743338919801464057}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\end{document}