Series expansions for the incomplete Lipschitz‐Hankel integralYe₀(a, z)

Abstract

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.

Numerical computation of incomplete Lipschitz‐Hankel integrals of the Hankel type for complex‐valued arguments

“…The first convergent factorial‐Neumann series expansion is obtained by combining the results of Mechaik and Dvorak [1995, equation (44)] and Mechaik and Dvorak [1996, equation (32)]. We have found that it is convenient to separate the integral into two parts, that is, an indefinite integral and the contribution due to the lower limit of integration, that is, In the above expression, we once again employ the principal branch of the natural logarithm, and the plus and minus signs correspond to the superscripts 1 and 2, respectively.…”

“…In addition, it was shown that the series expansions developed by Dvorak and Kuester [1990] are also valid for complex‐valued a and ς. This insight allowed for the development of series expansions for the computation of Ye 0 ( a , ς) for complex‐valued a and ς by Mechaik and Dvorak [1996].…”

“…However, once an ILHI or CILHI of zero order has been computed, then the higher integer‐order functions can be computed using the following recurrence relation: where and Z n = J n , Y n , H n (1) , or H n (2) . Note that a stability analysis of is required before it can be employed [ Dvorak and Kuester , 1990; Mechaik and Dvorak , 1996]. In addition, can also be applied to regular ILHIs since …”

“…Three of the series expansions in this paper are based on the results developed by Mechaik and Dvorak [1995, 1996], that is, the first convergent factorial‐Neumann series, the asymptotic factorial‐Neumann series, and the quasi‐Neumann series. While it was possible to remove the large exponential behavior from the CILHI representations for the first convergent factorial‐Neumann series and the asymptotic factorial Neumann series expansions, this was not possible in the case of the quasi‐Neumann series expansion.…”

“…While it was possible to remove the large exponential behavior from the CILHI representations for the first convergent factorial‐Neumann series and the asymptotic factorial Neumann series expansions, this was not possible in the case of the quasi‐Neumann series expansion. Therefore the quasi‐Neumann series representation for the CILHIs of the Hankel type cannot be employed over as large of a range in the complex a plane as when computing the ILHIs [e.g., Mechaik and Dvorak , 1996]. In order to make up for the limited range for this series, an additional series expansion, called the second convergent factorial‐Neumann series expansion, is developed.…”