H1 Hankel H1

H1.1 Introduction

Let x be a complex variable of ℂ \ {0, ∞} and let ν denote a parameter (independent of x).The function Hankel H1 (noted H_{ν}^{(1)}) 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. \\ (H1.1.1) \end{aligned}

Although 0 is a singularity of H1.1.1, the initial conditions can be given by

\begin{aligned}[x^{(−ν)}] H_{ν}^{(1)}(x) & = \frac{−iΓ(ν)}{\frac{π}{2^{ν}}}, \\ [x^{ν}] H_{ν}^{(1)}(x) & = \frac{−i e^{(−iπν)} Γ(−ν)}{π2^{ν}}. \end{aligned} ; (H1.1.2)

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

H1.2 Series and asymptotic expansions

H1.2.1 Asymptotic expansion at ∞. H1.2.1.1 First terms.

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

where

y_{0}(x) = \begin{matrix}\sqrt{2} & e & (− \frac{i}{4} π(2ν+1)) \\ √ & \overline{π} \end{matrix} − \frac{−(4ν^{2} − 1)\sqrt{2} e^{(− \frac{i}{4} π(2ν+1))} x}{8\sqrt{π} RootOf_{ξ,2}(1 + ξ^{2})}+

\frac{(4ν^{2} − 9)(4ν^{2} − 1)\sqrt{2} e^{(− \frac{i}{4} π(2ν+1))} x^{2}}{128\sqrt{π} RootOf_{ξ,2}(1 + ξ^{2})^{2}} −

\frac{−(4ν^{2} − 25)(4ν^{2} − 9)(4ν^{2} − 1)\sqrt{2} e^{(− \frac{i}{4} π(2ν+1))} x^{3}}{3072\sqrt{π} RootOf_{ξ,2}(1 + ξ^{2})^{3}} +

2 …

H1.2.1.2 General form.

H1.2.1.2.1 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) = \begin{matrix}\sqrt{2} & e & (− \frac{i}{4} π(2ν+1)) \\ √ & \overline{π} \end{matrix}

This recurrence has the closed form solution

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

(Γ(n + 1)8^{n}π^{\frac{3}{2}} ).

H1.2.2 Asymptotic expansion at 0.

H1.2.2.1 First terms.

H_{ν}^{(1)}(x) ≈ (\frac{−iΓ(ν)}{\frac{π}{2^{ν}}} − \frac{ix^{2}Γ(ν)}{\frac{4(ν−1)π}{2^{ν}}} − \frac{ix^{4}Γ(ν)}{\frac{32(ν−1)(ν−2)π}{2^{ν}}} −

\frac{ix^{6}Γ(ν)}{\frac{384(ν−1)(ν−2)(ν−3)π}{2^{ν}}} − \frac{ix^{8}Γ(ν)}{\frac{6144(ν−1)(ν−2)(ν−3)(ν−4)π}{2^{ν}}}

…)/x^{ν} + x^{ν}(\frac{−i e^{(−iπν)} Γ(−ν)}{π2^{ν}} + \frac{ix^{2} e^{(−iπν)} Γ(−ν)}{4(ν + 1)π2^{ν}}−

\frac{ix^{4} e^{(−iπν)} Γ(−ν)}{32(ν + 1)(ν + 2)π2^{ν}} + \frac{ix^{6} e^{(−iπν)} Γ(−ν)}{384(ν + 1)(ν + 2)(ν + 3)π2^{ν}} −

\frac{ix^{8} e^{(−iπν)} Γ(−ν)}{6144(ν + 1)(ν + 2)(ν + 3)(ν + 4)π2^{ν}} …).

(H1.2.2.1.1)

H1.2.2.2 General form. The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).