Bessel series expansions are derived for the incomplete Lipschitz‐Hankel integralJe0(a, z). These expansions are obtained by using contour integration techniques to evaluate the inverse Laplace transform representation for Je0(a, z). It is shown that one of the expansions can be used as a convergent series expansion for one definition of the branch cut and as an asymptotic expansion if the branch cut is chosen differently. The effects of the branch cuts are demonstrated by plotting the terms in the series for interesting special cases. The Laplace transform technique used in this paper simplifies the derivation of the series expansions, provides information about the resulting branch cuts, yields integral representations for Je0(a, z), and allows the series expansions to be extended to complex values of z. These series expansions can be used together with the expansions for Ye0(a, z), which are obtained in a separate paper, to compute numerous other special functions, encountered in electromagnetic applications. These include: incomplete Lipschitz‐Hankel integrals of the Hankel and modified Bessel form, incomplete cylindrical functions of Poisson form (incomplete Bessel, Struve, Hankel, and Macdonald functions), and incomplete Weber integrals (Lommel functions of two variables).

Transmission lines are standard wave guiding structures used for data transmission in multi-point communications. Signal attenuation, signal integriq, and transmission loss associated with transmission lines are modeled and investigated in this paper. Performance of transmitting data in coaxial cables and in strip transmission lines are analyzed using an accurate and efficient Method of Moments (MOM) solution based on solving Maxwell's equations for the electromagnetic fields associated with wave guiding structures supporting data transmission in finite homogeneous and isotropic media. The analysis in this paper models attenuation losses, conductive losses, and dielectric losses for a coaxial cable and a Printed Circuit Board (PCB). The models show that transmission loss is mainly dominated by conductive losses in metals for frequencies up to 25 GHz and by dielectric losses at much higher frequencies.Results show that high attenuation prevents data transmission at much higher frequencies unless the transmission line is designed using low-loss dielectric materials in the dielectric medium and high conductive material in the metals.

Three series expansions are derived for the incomplete Lipschitz‐Hankel integral YeO(a, z) for complex‐valued a and z. Two novel expansions are obtained by using contour integration techniques to evaluate the inverse Laplace transform representation for YeO(a, z). A third expansion is obtained by replacing the Neumann function by its Neumann series representation and integrating the resulting terms. An algorithm is outlined which chooses the most efficient expansion for given values of a and z. Comparisons of numerical results for these series expansions with those obtained by using numerical integration routines show that the expansions are very efficient and yield accurate results even for values of a and z for which numerical integration fails to converge. The integral representations for YeO(a, z) obtained in this paper are combined with previously obtained integral representations for Jeo(a, z) to derive integral representations for HeO(1) (a, z) and HeO(2) (α, z). Recurrence relations can be used to efficiently compute higher‐order incomplete Lipschitz‐Hankel integrals and to find integral representations and series expansions for these special functions and many other related functions.

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