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

Incomplete Lipschitz-Hankel integrals (ILHIs) form an important class of special functions since they appear in numerous applications in engineering and physics. While ILHIs of the Hankel type can be expressed as linear combinations of ILHIs of the Bessel and Neumann types, this procedure can lead to computational inaccuracies. As shown in this paper, these inaccuracies can be avoided by directly computing ILHIs of the Hankel types (both the first and second kinds). If desired, these results can then be combined to obtain accurate values for the ILHIs of the Bessel and Neumann types. Series representations for complementary incomplete Lipschitz-Hankel integrals (CILHIs) are also derived in this paper. CILHIs are often needed to avoid numerical inaccuracies caused by finite precision arithmetic. In order to better understand this group of special functions, the characteristics of ILHIs and CILHIs of the Hankel type are also discussed.