Hankel functions

2020 Mathematics Subject Classification: Primary: 33C10 [MSN][ZBL]

More commonly called Bessel functions (or Cylinder functions) of the third kind. These functions were introduced by H. Hankel in 1869.

They may be defined in terms of Bessel functions of the first and second kind (see Neumann function for the latter) as follows: \begin{align} &H^{(1)}_\nu = J_\nu + i Y_\nu\, ,\label{e:def_1}\\ &H^{(2)}_\nu = J_\nu - i Y_\nu\, .\label{e:def_2}\\ \end{align} $\nu$ is here a complex parameter. In particular, when $\nu\not\in \mathbb Z$, we have the expressions \begin{align} &H^{(1)}_\nu (z) = \frac{J_{-\nu} (z) - e^{-\nu \pi i} J_\nu (z)}{i\sin \nu\pi}\\ &H^{(2)}_\nu (z) = \frac{J_{-\nu} (z) - e^{\nu \pi i} J_\nu (z)}{-i\sin \nu\pi i}\, , \end{align} whereas for integer values $n$ of $\nu$ analogous formulas hold if we replace the right hand sides with their limits as $\nu\to n$.

This implies the important relations \begin{align*} & H^{(1}_{-\nu} (z) = e^{i\nu \pi} H^{(1)}_\nu (z)\\ &H^{(2)}_{-\nu} (z) = e^{-i\nu \pi} H^{(2)}_\nu (z)\, . \end{align*} When $\nu=p$ is real, the Bessel functions of the first kind take real values on the real axis. So it is obvious that, for $\nu = p$ real, $H^{(1)}_p$ and $H^{(2)}_p$ take complex conjugate values on the real axis. Moreover, \[ i^{p+1} H^{(1)}_p (ix) \qquad \mbox{and} \qquad i^{-(p+1)} H^{(2)}_p (-ix) \] are real if $x$ is real and positive.

Hankel functions have simple asymptotic formulas for large $|z|$ when $\nu=p$ is real: \begin{align*} &H^{(1)}_p (z) \sim \sqrt{\frac{2}{\pi z}} \exp \left(i \left(z - p \frac{\pi}{2} - \frac{\pi}{4}\right)\right)\, ,\\ &H^{(2)}_p (z) \sim \sqrt{\frac{2}{\pi z}} \exp \left(- i \left(z - p \frac{\pi}{2} - \frac{\pi}{4}\right)\right)\, . \end{align*} The Hankel functions of half-integral $p = n +\frac{1}{2}$, $n\in \mathbb Z$, can be expressed in terms of elementary functions, in particular: \begin{align*} &H^{(1)}_{1/2} (z) = \sqrt{\frac{2}{\pi z}} \frac{e^{iz}}{i}\, ,\\ &H^{(2)}_{1/2} (z) = -\sqrt{\frac{2}{\pi z}} \frac{e^{-iz}}{i}\, . \end{align*} See Cylinder functions for additional references.