H2 Hankel H2

H2.1 Introduction

Let $x$ be a complex variable of $\mathbb{C} \setminus \{0,\infty\}$ and let $\nu$ denote a parameter (independent of $x$ ).The function Hankel H2 (noted $\operatorname{H} _{\nu} ^{(2)}$ ) is defined by the following second order differential equation

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

H2.1.1

Although $0$ is a singularity of H2.1.1, the initial conditions can be given by

$\begin{equation*} \begin{split} \Bigl[x^{(-\nu)}\Bigr] \operatorname{H} _{\nu} ^{(2)} (x)& =\frac{i \Gamma (\nu)}{\frac{\pi}{2^{\nu}}}, \\ \bigl[x^{\nu}\bigr] \operatorname{H} _{\nu} ^{(2)} (x)& =\frac{i \operatorname{e} ^{i \pi \nu} \Gamma (-\nu)}{\pi 2^{\nu}}. \end{split} \end{equation*}$

H2.1.2

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

Related functions: Hankel H1,Bessel Y,Bessel J

H2.2 Series and asymptotic expansions

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H2.2.1 Asymptotic expansion at $\infty$

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H2.2.1.1 First terms

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

where

$\begin{equation*} \begin{split} y _{0} (x)& =\frac{\sqrt{2} \operatorname{e} ^{\frac{i}{4} \pi (2 \nu + 1)}}{\sqrt{\pi}} - \frac{-\bigl(4 \nu^{2} - 1\bigr) \sqrt{2} \operatorname{e} ^{\frac{i}{4} \pi (2 \nu + 1)} x}{8 \sqrt{\pi} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)} + \\ & \quad{}\quad{}\frac{\bigl(4 \nu^{2} - 9\bigr) \bigl(4 \nu^{2} - 1\bigr) \sqrt{2} \operatorname{e} ^{\frac{i}{4} \pi (2 \nu + 1)} x^{2}}{128 \sqrt{\pi} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{2}} - \\ & \quad{}\quad{}\frac{-\bigl(4 \nu^{2} - 25\bigr) \bigl(4 \nu^{2} - 9\bigr) \bigl(4 \nu^{2} - 1\bigr) \sqrt{2} \operatorname{e} ^{\frac{i}{4} \pi (2 \nu + 1)} x^{3}}{3072 \sqrt{\pi} \operatorname{RootOf} _{\xi,1} \bigl(1 + \xi^{2}\bigr)^{3}} + 2 \ldots \end{split} \end{equation*}$

H2.2.1.2 General form

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H2.2.1.2.1 Auxiliary function $y _{0} (x)$

The coefficients $u (n)$

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

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

whose initial conditions are given by

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

This recurrence has the closed form solution

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

H2.2.2 Asymptotic expansion at

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H2.2.2.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{H} _{\nu} ^{(2)} (x)\approx \Biggl(\frac{i \Gamma (\nu)}{\frac{\pi}{2^{\nu}}} + \frac{i x^{2} \Gamma (\nu)}{\frac{4 (\nu - 1) \pi}{2^{\nu}}} + \frac{i x^{4} \Gamma (\nu)}{\frac{32 (\nu - 1) (\nu - 2) \pi}{2^{\nu}}} + \\ & \quad{}\quad{}\frac{i x^{6} \Gamma (\nu)}{\frac{384 (\nu - 1) (\nu - 2) (\nu - 3) \pi}{2^{\nu}}} + \frac{i x^{8} \Gamma (\nu)}{\frac{6144 (\nu - 1) (\nu - 2) (\nu - 3) (\nu - 4) \pi}{2^{\nu}}} \\ & \quad{}\quad{}\ldots\Biggr)\Bigg/x^{\nu} + x^{\nu} \Biggl(\frac{i \operatorname{e} ^{i \pi \nu} \Gamma (-\nu)}{\pi 2^{\nu}} - \frac{i x^{2} \operatorname{e} ^{i \pi \nu} \Gamma (-\nu)}{4 (\nu + 1) \pi 2^{\nu}} + \\ & \quad{}\quad{}\frac{i x^{4} \operatorname{e} ^{i \pi \nu} \Gamma (-\nu)}{32 (\nu + 1) (\nu + 2) \pi 2^{\nu}} - \frac{i x^{6} \operatorname{e} ^{i \pi \nu} \Gamma (-\nu)}{384 (\nu + 1) (\nu + 2) (\nu + 3) \pi 2^{\nu}} + \\ & \quad{}\quad{}\frac{i x^{8} \operatorname{e} ^{i \pi \nu} \Gamma (-\nu)}{6144 (\nu + 1) (\nu + 2) (\nu + 3) (\nu + 4) \pi 2^{\nu}}\ldots\Biggr). \end{split} \end{equation*}$

H2.2.2.1.1

H2.2.2.2 General form

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