BSI Bessel I

BSI.1 Introduction

Let x be a complex variable of ℂ \ {0, ∞} and let ν denote a parameter (independent of x).The function Bessel I (noted I_{ν}) is defined by the following second order differential equation

\begin{aligned}& −x^{2} + ν^{2}y(x) + x\frac{∂y(x)}{∂x} + x^{2} \frac{∂^{2}y(x)}{∂x^{2}} & = 0. \\ (BSI.1.1) \end{aligned}

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

\begin{aligned}& \frac{∂ \frac{I_{ν}(x)}{x^{ν}}}{∂x} & = \frac{1}{Γ(ν + 1)2^{ν}} . \\ (BSI.1.2) \end{aligned}

The formulae of this document are valid for −2ν ∉ ℤ. Related function: Bessel K

BSI.2 Series and asymptotic expansions

BSI.2.1 Asymptotic expansion at ∞. BSI.2.1.1 First terms.

I_{ν}(x) ≈ e^{(− \frac{1}{x})} \sqrt{x}y_{0}(x),

where

\begin{aligned}y_{0}(x) & = \frac{\sqrt{2}}{2\sqrt{π}} − \frac{−(4ν^{2} − 1)\sqrt{2}x}{16\sqrt{π}} + \frac{(4ν^{2} − 9)(4ν^{2} − 1)\sqrt{2}x^{2}}{256\sqrt{π}} − \\ & \frac{−(4ν^{2} − 25)(4ν^{2} − 9)(4ν^{2} − 1)\sqrt{2}x^{3}}{6144\sqrt{π}} + 2 … \end{aligned}

BSI.2.1.2 General form. BSI.2.1.2.1 Auxiliary function y_{0}(x). The coefficients u(n) of y_{0}(x) satisfy the

following recurrence

8u(n)n + u(n − 1)(−4ν^{2} − 3 + 4n + 4(n − 1)^{2}) = 0

whose initial conditions are given by

u(0) = \frac{\sqrt{2}}{2\sqrt{π}}

This recurrence has the closed form solution

u(n) = \frac{2^{(n+ \frac{1}{2})}Γ(n + ν + \frac{1}{2})Γ(n − ν + \frac{1}{2}) sin(\frac{π(2ν+1)}{2})(−2)^{n}}{28^{n}π^{\frac{3}{2}} Γ(n + 1)} .

2 BSI BESSEL I

BSI.2.2 Asymptotic expansion at 0.

BSI.2.2.1 First terms.

I_{ν}(x) ≈ x^{ν}(\frac{1}{Γ(ν + 1)2^{ν}} + \frac{x^{2}}{4Γ(ν + 1)2^{ν}(ν + 1)}+

\frac{x^{4}}{32Γ(ν + 1)2^{ν}(ν + 1)(ν + 2)}+

\frac{x^{6}}{384Γ(ν + 1)2^{ν}(ν + 1)(ν + 2)(ν + 3)} …).

(BSI.2.2.1.1)

BSI.2.2.2 General form.

\begin{aligned}& I_{ν}(x) ≈ x^{ν} ∑_{n=0}^{∞} u(n)x^{n}. \\ (BSI.2.2.2.1) \end{aligned}

The coefficients u(n) satisfy the recurrence

\begin{aligned}& u(n)(−ν^{2} + (ν + n)^{2}) − u(n − 2) & = 0. \\ (BSI.2.2.2.2) \end{aligned}

Initial conditions of BSI.2.2.2.2 are given by

\begin{aligned}u(0) & = \frac{1}{Γ(ν + 1)2^{ν}} , \\ u(1) & = 0. \end{aligned} ; (BSI.2.2.2.3)

The recurrence BSI.2.2.2.2 has the closed form solution

\begin{aligned}u(2n) & = \frac{1}{2^{ν}4^{n}Γ(n + 1)Γ(n + ν + 1)}, \\ u(2n + 1) & = 0. \end{aligned} ; (BSI.2.2.2.4)