BSJ Bessel J

BSJ.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 J (noted $\operatorname{J} _{\nu}$ ) is defined by the following second order differential equation

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

BSJ.1.1

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

$\begin{equation*} \begin{split} \frac{\partial \frac{\operatorname{J} _{\nu} (x)}{x^{\nu}}}{\partial x}& =\frac{1}{\Gamma (\nu + 1) 2^{\nu}}. \end{split} \end{equation*}$

BSJ.1.2

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

Related functions: Hankel H1,Hankel H2,Bessel Y

BSJ.2 Series and asymptotic expansions

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BSJ.2.1 Asymptotic expansion at

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BSJ.2.1.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{J} _{\nu} (x)\approx x^{\nu} \Biggl(\frac{1}{\Gamma (\nu + 1) 2^{\nu}} - \frac{x^{2}}{4 \Gamma (\nu + 1) 2^{\nu} (\nu + 1)} + \\ & \quad{}\quad{}\frac{x^{4}}{32 \Gamma (\nu + 1) 2^{\nu} (\nu + 1) (\nu + 2)} - \\ & \quad{}\quad{}\frac{x^{6}}{384 \Gamma (\nu + 1) 2^{\nu} (\nu + 1) (\nu + 2) (\nu + 3)}\ldots\Biggr). \end{split} \end{equation*}$

BSJ.2.1.1.1

BSJ.2.1.2 General form

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$\begin{equation*} \begin{split} & \operatorname{J} _{\nu} (x)\approx x^{\nu} \sum_{n = 0}^{\infty} u (n) x^{n}. \end{split} \end{equation*}$

BSJ.2.1.2.1

The coefficients $u (n)$

satisfy the recurrence

$\begin{equation*} \begin{split} u (n) \bigl(-\nu^{2} + (\nu + n)^{2}\bigr) + u (n - 2)& =0. \end{split} \end{equation*}$

BSJ.2.1.2.2

Initial conditions of BSJ.2.1.2.2 are given by

$\begin{equation*} \begin{split} u (1)& =0, \\ u (0)& =\frac{1}{\Gamma (\nu + 1) 2^{\nu}}. \end{split} \end{equation*}$

BSJ.2.1.2.3

The recurrence BSJ.2.1.2.2 has the closed form solution

$\begin{equation*} \begin{split} u (2 n + 1)& =0, \\ u (2 n)& =\frac{(-1)^{n}}{2^{\nu} 4^{n} \Gamma (n + 1) \Gamma (n + \nu + 1)}. \end{split} \end{equation*}$

BSJ.2.1.2.4

BSJ.2.2 Asymptotic expansion at $\infty$

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BSJ.2.2.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{J} _{\nu} (x)\approx \operatorname{e} ^{\biggl(-\frac{\operatorname{RootOf} _{\xi,2} (1 + \xi^{2})}{x}\biggr)} \sqrt{x} y _{0} (x) + \\ & \quad{}\quad{}\operatorname{e} ^{\biggl(-\frac{\operatorname{RootOf} _{\xi,1} (1 + \xi^{2})}{x}\biggr)} \sqrt{x} y _{1} (x), \end{split} \end{equation*}$

where

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

BSJ.2.2.2 General form

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BSJ.2.2.2.1 Auxiliary function $y _{0} (x)$

The coefficients $u (n)$

of $y _{0} (x)$ satisfy the following recurrence

$\begin{equation*} \begin{split} & 8 u (n) n \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr) + u (n - 1) \bigl(-4\nu^{2} - 3 + 4 n + 4 (n - 1)^{2}\bigr)=0 \end{split} \end{equation*}$

whose initial conditions are given by

$\begin{equation*} \begin{split} u (0)& =\frac{\sqrt{2} \operatorname{e} ^{\bigl(-\frac{i}{4}\pi (2 \nu + 1)\bigr)}}{2 \sqrt{\pi}} \end{split} \end{equation*}$

This recurrence has the closed form solution

$\begin{equation*} \begin{split} u (n)& =\Biggl(2^{\Bigl(n + \frac{1}{2}\Bigr)} \operatorname{RootOf} _{\xi,2} \bigl(1 + \xi^{2}\bigr)^{n} \Gamma \biggl(n - \nu + \frac{1}{2}\biggr) \Gamma \biggl(\nu + \frac{1}{2} + n\biggr) \\ & \quad{}\quad{}\operatorname{sin} \Biggl(\frac{\pi (2 \nu + 1)}{2}\Biggr) (-2)^{n} (-1)^{n} \operatorname{e} ^{\Bigl(-\frac{i}{4}\pi (2 \nu + 1)\Bigr)}\Biggr)\Bigg/ \\ & \quad{}\quad{}\Biggl(2 \Gamma (n + 1) 8^{n} \pi^{\frac{3}{2}}\Biggr). \end{split} \end{equation*}$

BSJ.2.2.2.2 Auxiliary function $y _{1} (x)$

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

$\begin{equation*} \begin{split} u (0)& =\frac{\sqrt{2} \operatorname{e} ^{\frac{i}{4} \pi (2 \nu + 1)}}{2 \sqrt{\pi}} \end{split} \end{equation*}$

This recurrence has the closed form solution

$\begin{equation*} \begin{split} u (n)& =\Biggl(2^{\Bigl(n + \frac{1}{2}\Bigr)} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{n} \Gamma \biggl(n - \nu + \frac{1}{2}\biggr) \Gamma \biggl(\nu + \frac{1}{2} + n\biggr) \\ & \quad{}\quad{}\operatorname{sin} \Biggl(\frac{\pi (2 \nu + 1)}{2}\Biggr) (-2)^{n} (-1)^{n} \operatorname{e} ^{\frac{i}{4} \pi (2 \nu + 1)}\Biggr)\Bigg/ \\ & \quad{}\quad{}\Biggl(2 \Gamma (n + 1) 8^{n} \pi^{\frac{3}{2}}\Biggr). \end{split} \end{equation*}$