Hankel function of the second kind is used for forward propagating (+z-axis) wave, and Hankel function of the first kind
is used for reverse propagating (-z-axis) wave .
where the [h.sup.(1)'.sub.n] is the derivative of the spherical Hankel function of the first kind
(see Chapter 10 in ).
[H.sup.(1).sub.0] (x) is the Hankel function of the first kind
and zero order, [C.sub.1] is the integration contour along the banks of the cut of the function v = [square root of ([k.sup.2] - [w.sup.2])] that is a parallel line to the imaginary axis upwards from the branch point.
where [H.sup.(1).sub.0] = [J.sub.0] + i[Y.sub.0] is the Hankel function of the first kind
and zero order (19, 20), [J.sub.0] and [Y.sub.0] are the corresponding Bessel functions of the first and second kind, and [beta] is the same as in Eq.