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) [19], 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 [32], 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 [20] 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 [7].

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 can exist.