BSK Bessel K

BSK.1 Introduction

Let x be a complex variable of ℂ \ {0, ∞} and let ν denote a parameter (independent of x).The function Bessel K (noted K_{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. \\ (BSK.1.1) \end{aligned}

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

\begin{aligned}[x^{(−ν)}] K_{ν}(x) & = 2^{(ν−1)}Γ(ν), \\ [x^{ν}] K_{ν}(x) & = −\frac{π}{Γ(μ + 1) sin(μπ)2^{(μ+1)}} . \end{aligned} (BSK.1.2)

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

BSK.2 Series and asymptotic expansions

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

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

where

y_{0}(x) = \frac{\sqrt{2}\sqrt{π}}{2} − \frac{−(4ν^{2} − 1)\sqrt{2}\sqrt{π}x}{16}+

\frac{(4ν^{2} − 9)(4ν^{2} − 1)\sqrt{2}\sqrt{π}x^{2}}{256} −

\frac{−(4ν^{2} − 25)(4ν^{2} − 9)(4ν^{2} − 1)\sqrt{2}\sqrt{π}x^{3}}{6144} + 2 …

BSK.2.1.2 General form. BSK.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}\sqrt{π}}{2}

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}\sqrt{π}Γ(n + 1)} .

BSK.2.2 Asymptotic expansion at 0.

BSK.2.2.1 First terms.

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

\frac{x^{6}2^{(ν−1)}Γ(ν)}{384(ν − 1)(ν − 2)(ν − 3)} + \frac{x^{8}2^{(ν−1)}Γ(ν)}{6144(ν − 1)(ν − 2)(ν − 3)(ν − 4)} …)/

x^{ν} + x^{ν}(−\frac{π}{Γ(μ + 1) sin(μπ)2^{(μ+1)}} −

\frac{x^{2}π}{4(ν + 1)Γ(μ + 1) sin(μπ)2^{(μ+1)}} −

\frac{x^{4}π}{32(ν + 1)(ν + 2)Γ(μ + 1) sin(μπ)2^{(μ+1)}} −

\frac{x^{6}π}{384(ν + 1)(ν + 2)(ν + 3)Γ(μ + 1) sin(μπ)2^{(μ+1)}} −

\frac{x^{8}π}{6144(ν + 1)(ν + 2)(ν + 3)(ν + 4)Γ(μ + 1) sin(μπ)2^{(μ+1)}} …).

(BSK.2.2.1.1)

BSK.2.2.2 General form. The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).