Series expansions for the incomplete Lipschitz‐Hankel integral<i>Ye</i>₀(a, <i>z</i>)

Abstract

Three series expansions are derived for the incomplete Lipschitz‐Hankel integral Ye O (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 Ye O (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 Ye O (a, z ) obtained in this paper are combined with previously obtained integral representations for Je o (a, z ) to derive integral representations for He O (1) (a, z ) and He O (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.