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\begin{document}
\chapter*{H2 Hankel H2}
\label{H2}
\section*{H2.1 Introduction}
\label{H2:intro}


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*}
\label{H2:diffeq}
\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}\tag{H2.1.1}
\end{equation*}



Although $0$ is a singularity of H2.1.1, the initial conditions can be given by 
\begin{equation*}
\label{H2:inicond}
\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}\tag{H2.1.2}
\end{equation*}



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


Related functions: \href{http://algo.inria.fr/esf/function/H1/H1.html#H1}{Hankel H1},\href{http://algo.inria.fr/esf/function/BSY/BSY.html#BSY}{Bessel Y},\href{http://algo.inria.fr/esf/function/BSJ/BSJ.html#BSJ}{Bessel J}
\section*{H2.2 Series and asymptotic expansions}
\label{H2:asympt}
\subsection*{H2.2.1 Asymptotic expansion at $\infty$}
\label{743573170919915183}
\subsubsection*{H2.2.1.1 First terms}
\label{743573943785527953}



\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*}

\subsubsection*{H2.2.1.2 General form}
\label{743573983954069046}
\paragraph*{H2.2.1.2.1 Auxiliary function $y _{0} (x)$}
\label{743573517937362402}
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*}
\subsection*{H2.2.2 Asymptotic expansion at $0$}
\label{743572989692312125}
\subsubsection*{H2.2.2.1 First terms}
\label{H2:asympt:0:termsec}



\begin{equation*}
\label{H2:asympt:0:terms}
\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}\tag{H2.2.2.1.1}
\end{equation*}

\subsubsection*{H2.2.2.2 General form}
\label{743572236449634754}
The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).\end{document}