BSK Bessel K

BSK.1 Introduction

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

$\begin{equation*} \begin{split} -x^{2} + \nu^{2}y (x) + x \frac{\partial y (x)}{\partial x} + x^{2} \frac{\partial^{2} y (x)}{\partial x^{2}}& =0. \end{split} \end{equation*}$

BSK.1.1

Although $0$ is a singularity of BSK.1.1, the initial conditions can be given by

$\begin{equation*} \begin{split} \Bigl[x^{(-\nu)}\Bigr] \operatorname{K} _{\nu} (x)& =2^{(\nu - 1)} \Gamma (\nu), \\ \bigl[x^{\nu}\bigr] \operatorname{K} _{\nu} (x)& =-\frac{\pi}{\Gamma (\mu + 1) \operatorname{sin} (\mu \pi) 2^{(\mu + 1)}}. \end{split} \end{equation*}$

BSK.1.2

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

Related function: Bessel I

BSK.2 Series and asymptotic expansions

top up back next into bottom

BSK.2.1 Asymptotic expansion at $\infty$

top up back next into bottom

BSK.2.1.1 First terms

top up back next into bottom

$\begin{equation*} \begin{split} & \operatorname{K} _{\nu} (x)\approx \operatorname{e} ^{\Bigl(-\frac{1}{x}\Bigr)} \sqrt{x} y _{0} (x), \end{split} \end{equation*}$

where

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

BSK.2.1.2 General form

top up back next into bottom

BSK.2.1.2.1 Auxiliary function $y _{0} (x)$

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

$\begin{equation*} \begin{split} u (n)& =\frac{2^{\bigl(n + \frac{1}{2}\bigr)} \Gamma \Bigl(n + \nu + \frac{1}{2}\Bigr) \Gamma \Bigl(n - \nu + \frac{1}{2}\Bigr) \operatorname{sin} \biggl(\frac{\pi (2 \nu + 1)}{2}\biggr) (-2)^{n}}{2 8^{n} \sqrt{\pi} \Gamma (n + 1)}. \end{split} \end{equation*}$

BSK.2.2 Asymptotic expansion at

top up back next into bottom

BSK.2.2.1 First terms

top up back next into bottom

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

BSK.2.2.1.1

BSK.2.2.2 General form

top up back next into bottom

The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).