Hunkel functions
Bessel function of third kind
\begin{align*}
H_{\nu}^{(1)} (x) &= J_{\nu} (x) + {\bf j}\,Y_{\nu} (x) = \frac{J_{-\nu} (x) - e^{-\nu\pi{\bf j} J_{\nu} (x)} }{{\bf j}\,\sin \nu\pi} ,
\\
H_{\nu}^{(2)} (x) &= J_{\nu} (x) - {\bf j}\,Y_{\nu} (x) = \frac{J_{-\nu} (x) - e^{\nu\pi{\bf j} J_{\nu} (x)} }{-{\bf j}\,\sin \nu\pi} ,
\end{align*}
where j is a unit vector in positive vertical direction on the complex plane ℂ so that j² = -1.
- Bowman, Frank Introduction to Bessel Functions (Dover: New York, 1958). ISBN 0-486-60462-4.
- Dutka, J., On the early history of Bessel functions, Archive for History of Exact Sciences, volume 49, pages 105–134 (1995). https://doi.org/10.1007/BF00376544
- Watson, G.N., A Treatise on the Theory of Bessel Functions,
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