Bessel series expansions are derived for the incomplete Lipschitz-Hankel integral Je0(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). ¿ American Geophysical Union 1995