where a is the cylinder's radius, [J.sub.n](x) is the Bessel functions of first kind, [H.sup.(2).sub.n](x) is the Hankel functions
of second kind, [J'.sub.n](x) is the derivative of [J.sub.n](x), [H'.sup.(2).sub.n](x) is the derivative of [H.sup.(2).sub.n](x) , n is the azimuthal mode order, k is the wave-number, and [eta] is the intrinsic impedance.
The ordinary hypergeometric [19-25], confluent hypergeometric [26-29], Coulomb wave functions [30,31], Bessel and Hankel functions
, incomplete Beta and Gamma functions [33-35], Hermite functions [36,37], Goursat and Appell generalized hypergeometric functions of two variables of the first kind [38,39], and other known special functions have been used as expansion functions.
There are several choices for the two independent functions: Bessel [J.sub.m]([k.sub.i]) and Neumann [N.sub.m] functions or Hankel functions
of the first [H.sup.(1).sub.m] and the second kind [H.sup.(2).sub.m].
where [J.sub.m] and [H.sup.(l).sub.m] denote Bessel functions and Hankel functions
of first kind of order m, respectively, while [k.sub.0] is the free-space wave number.
Here A, B, C, and D stand for arbitrary constants, [J.sub.0], [J.sub.1] and [Y.sub.0], [Y.sub.1] are Bessel and Hankel functions
of orders zero and one correspondingly.
Furthermore, Bessel functions are transformed using Hankel functions
and the reflection formula:
On the other hand, when [b.sub.s] [right arrow] 0, thereby Hankel functions
[H.sup.(1).sub.n]'([[beta].sub.p][b.sub.s]) [right arrow] 0 and [H.sup.(2).sub.n]'([[beta].sub.p][b.sub.s]) [right arrow] 0; 15) turns into
In the previous expressions [H.sup.(2).sub.n] ([k.sub.[rho]m] [absolute value of [rho] - [rho]l]) are Hankel functions
of second kind and
Using Equations (20)-(21), the identities  and the following recursive relation when differentiating the Bessel and Hankel functions
the ith, jth derivatives for the matrix entries can be derived
By assuming that [beta]r >> 1 (this entails that the observation domain is only at few wavelengths from the source domain) and by resorting to the asymptotic form of the Hankel functions
for large argument, the relationship [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCI.] holds .
where [H.sup.(1).sub.n] = [J.sub.n] + [iY.sub.n] are Hankel functions
of the first kind and nth order.
The general solution is a linear combination of radially outward and inward propagating waves expressed as Hankel functions
. To avoid the singularity at the axis, [Rho] = 0, only the real part of the associated Hankel function