in the immediate vicinity of this point, where . Here, is a small real constant. We introduce at this point principally as a mathematical artifice to ensure that remains single-valued and finite. However, as will become clear later on, has a physical significance in terms of damping or spontaneous excitation.

In the immediate vicinity of the resonance point, , Eqs. (555) and (581)
yield

(583) |

Let

(584) | |||

(585) |

In the limit , Eq. (582) transforms into

(586) |

where and are two arbitrary constants.

Let

(588) | |||

(589) |

where

(590) |

(591) |

where and are two arbitrary constants.

Now, the Bessel functions , , , and are all perfectly
well-defined for
complex arguments, so the two expressions (587) and (592) must, in fact, be
*identical*.
In particular, the constants and must somehow be related to the
constants and . In order to establish this relationship, it
is convenient to investigate the behaviour of the expressions
(587) and (592) in the limit of small : *i.e.*,
.
In this limit,

where is Euler's constant, and is assumed to lie on the positive real -axis. It follows, by a comparison of Eqs. (587), (592), and (593)-(596), that the choice

ensures that the expressions (587) and (592) are indeed identical.

Now, in the limit
,

(599) | |||

(600) |

where is assumed to lie on the negative real -axis. It is clear that the solution is unphysical, since it blows up in the evanescent region . Thus, the coefficient in expression (592) must be set to zero in order to prevent from blowing up as . According to Eq. (597), this constraint implies that

In the limit
,

where is assumed to lie on the positive real -axis. It follows from Eqs. (587), (601), and (602)-(603) that in the non-evanescent region () the most general

where is an arbitrary constant.

Suppose that a plane electromagnetic wave, polarized in the
-direction, is launched
from an antenna, located at large positive , towards the resonance
point at .
It is assumed that at the launch point.
In the non-evanescent region, , the wave can be
represented as a linear combination
of propagating WKB solutions:

(605) |

A comparison of Eqs. (604) and (606) shows that if then . In other words, there is a reflected wave, but no incident wave. This corresponds to the