BSY Bessel Y

BSY.1 Introduction

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

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

[x^{(−ν)}] Y_{ν}(x) = − \frac{1}{\frac{Γ(−ν+1) sin(νπ)}{2^{ν}}} ,

[x^{ν}] Y_{ν}(x) = \frac{cos(νπ)}{Γ(ν + 1)2^{ν} sin(νπ)}.

(BSY.1.2)

The formulae of this document are valid for −2ν ∉ ℤ. Related functions: Hankel H1,Hankel H2,Bessel J

BSY.2 Series and asymptotic expansions

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

Y_{ν}(x) ≈ e^{(− \frac{RootOf_{ξ,2}(1+ξ^{2})}{x})} \sqrt{x}y_{0}(x)+

e^{(− \frac{RootOf_{ξ,1}(1+ξ^{2})}{x})} \sqrt{x}y_{1}(x),

where

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

\frac{i(4ν^{2} − 9)(4ν^{2} − 1)\sqrt{2} e^{(− \frac{i}{4} π(2ν+1))} x^{2}}{256\sqrt{π} RootOf_{ξ,2}(1 + ξ^{2})^{2}} +

\frac{−i(4ν^{2} − 25)(4ν^{2} − 9)(4ν^{2} − 1)\sqrt{2} e^{(− \frac{i}{4} π(2ν+1))} x^{3}}{6144\sqrt{π} RootOf_{ξ,2}(1 + ξ^{2})^{3}} +

2 …

y_{1}(x) = \frac{i\sqrt{2} e^{\frac{i}{4} π(2ν+1)}}{2\sqrt{π}} − \frac{−i(4ν^{2} − 1)\sqrt{2} e^{\frac{i}{4} π(2ν+1)} x}{16\sqrt{π} RootOf_{ξ,1}(1 + ξ^{2})}+

\frac{i(4ν^{2} − 9)(4ν^{2} − 1)\sqrt{2} e^{\frac{i}{4} π(2ν+1)} x^{2}}{256\sqrt{π} RootOf_{ξ,1}(1 + ξ^{2})^{2}} −

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

BSY.2.1.2 General form.

BSY.2.1.2.1 Auxiliary function y_{0}(x). The coefficients u(n) of y_{0}(x) satisfy the following recurrence

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

whose initial conditions are given by

u(0) = \frac{−i\sqrt{2} e^{(− \frac{i}{4} π(2ν+1))}}{2\sqrt{π}}

This recurrence has the closed form solution

u(n) = (−i2^{(n+ \frac{1}{2})} RootOf_{ξ,2}(1 + ξ^{2})^{n}Γ(ν + \frac{1}{2} + n)Γ(n − ν + \frac{1}{2})

sin(\frac{π(2ν + 1)}{2})(−2)^{n}(−1)^{n} e^{(− \frac{i}{4} π(2ν+1))})/

(28^{n}π^{\frac{3}{2}} Γ(n + 1)).

BSY.2.1.2.2 Auxiliary function y_{1}(x). The coefficients u(n) of y_{1}(x) satisfy the following recurrence

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

whose initial conditions are given by

u(0) = \frac{i\sqrt{2} e^{\frac{i}{4} π(2ν+1)}}{2\sqrt{π}}

This recurrence has the closed form solution

u(n) = (i2^{(n+ \frac{1}{2})} RootOf_{ξ,1}(1 + ξ^{2})^{n}Γ(ν + \frac{1}{2} + n)Γ(n − ν + \frac{1}{2})

sin(\frac{π(2ν + 1)}{2})(−2)^{n}(−1)^{n} e^{\frac{i}{4} π(2ν+1)})/

(28^{n}π^{\frac{3}{2}} Γ(n + 1)).

BSY.2.2 Asymptotic expansion at 0.

BSY.2.2.1 First terms.

Y_{ν}(x) ≈ (− \frac{1}{\frac{Γ(−ν+1) sin(νπ)}{2^{ν}}} − \frac{x^{2}}{\frac{4(ν−1)Γ(−ν+1) sin(νπ)}{2^{ν}}} −

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

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

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

x^{ν}(\frac{cos(νπ)}{Γ(ν + 1)2^{ν} sin(νπ)} − \frac{x^{2} cos(νπ)}{4(ν + 1)Γ(ν + 1)2^{ν} sin(νπ)}

+ \frac{x^{4} cos(νπ)}{32(ν + 1)(ν + 2)Γ(ν + 1)2^{ν} sin(νπ)}−

\frac{x^{6} cos(νπ)}{384(ν + 1)(ν + 2)(ν + 3)Γ(ν + 1)2^{ν} sin(νπ)}+

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

(BSY.2.2.1.1)

BSY.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).