## 1. Introduction

[2] Incomplete Lipschitz-Hankel integrals (ILHIs) appear in numerous applications in engineering and physics. *Agrest and Maksimov* [1971], *Dvorak and Kuester* [1990], *Dvorak* [1994a], and *Mechaik and Dvorak* [1995] describe a number of these applications and provide references to a large body of literature on the subject. ILHIs can be used to obtain the analytical solutions in many canonical problems, for example, transient plane waves obliquely incident on a conductive half space [*Pao et al.*, 1996a, 1996b], the time domain surface impedance of a lossy half space [*Pao et al.*, 2004], scattering by a 2-D conducting strip [*Dvorak*, 1994b], plane wave diffraction by a 2-D aperture in a ground plane [*Dvorak and Pao*, 2005], lossy transmission line modeling [*Zhou et al.*, 2003], full wave interconnect modeling [*Heckmann and Dvorak*, 1999; *Kabir et al.*, 2001], etc. Some of these applications involved ILHIs of the Bessel type, and some employed ILHIs of the Hankel type.

### 1.1. Previous Work

[3] The general ILHI was defined by *Agrest and Maksimov* [1971] as

where *Z*_{ν}(*t*) is one of the Bessel or Hankel functions. Series expansions were developed by *Dvorak and Kuester* [1990] for the computation of *Je*_{0} (*a*, ς) for complex-valued *a* and real-valued ς. Two of the series expansions developed by *Dvorak and Kuester* [1990] were derived using recurrence relations. Later, Laplace transforms were employed by *Mechaik and Dvorak* [1995] to gain more insight into the development of the series expansions for *Je*_{0} (*a*, ς). 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].

[4] In the formulation of many problems [e.g., *Dvorak and Kuester*, 1990; *Heckmann and Dvorak*, 1999], it is often found that when ∣ς∣ is large, it can be beneficial to employ complementary incomplete Lipschitz-Hankel integrals (CILHIs) instead of regular ILHIs. The CILHIs are defined as

where the lower limit of integration, δ, is chosen so that the integral converges at infinity in the complex *t* plane so that *Ze*_{ν} (*a*, δ, 0) has an analytical solution. Using the definitions in (1) and (2), one can express the CILHIs in terms of regular ILHIs, that is,

### 1.2. Current Work

[5] Theoretically, ILHIs of the Hankel type can be obtained via *Je*_{0} (*a*, ς) and *Ye*_{0} (*a*, ς), that is,

Unfortunately, the use of (4) can lead to round-off errors when ∣Im(ς)∣ is large. The cause of these round-off errors is discussed in section 3. Fortunately, we have found that accurate results can be obtained by directly computing the ILHIs of Hankel form. The associated ILHIs of Bessel and Neumann types can then be obtained via

[6] To uniquely define the CILHIs of Hankel type, we specify the lower limit of integration in (2) as

which is valid for ν < 1/2. In (4) and (7), and throughout this paper, the upper and lower signs correspond to the CILHIs or ILHIs associated with Hankel functions of the first and second kinds, respectively. Since Hankel functions possess branch cuts along the negative real-*t* axis, δ is chosen on opposite sides of the branch cut for the Hankel functions of the first and second kinds when Re(*a*) < 0, as in (7), so that the integrand of (2) converges at δ. An example is given in section 12 to illustrate the advantage of using CILHIs in a physical problem formulation. Because of the computational advantages offered by the CILHIs, series expansions are also developed for these functions in this paper.

[7] This paper concentrates on the computation of ILHIs and CILHIs of zero order since they are encountered most often in applications. 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 (8) is required before it can be employed [*Dvorak and Kuester*, 1990; *Mechaik and Dvorak*, 1996]. In addition, (8) can also be applied to regular ILHIs since

[8] 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. 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. This series also provides an additional choice for computing the ILHIs with better accuracy and efficiency. Furthermore, since the first convergent factorial-Neumann series does not converge well when *a* approaches zero, a Struve function series expansion is also derived to provide better efficiency and accuracy for very small ∣*a*ς∣. With the addition of these two new series representations, it is now possible to accurately and efficiently compute CILHIs and ILHIs of the Hankel type for all complex *a* and ς.

[9] While we will assume that Re(ς) ≥ 0 when developing the series expansions for the ILHIs and CILHs, other values of ς can be handled by making use of the following identities [*Watson*, 1944, p. 75, equations (5) and (6)]:

where ψ = arg(ς). If (11) and (12) are employed together with (1), then it can be shown that

Furthermore, by employing (13) and (14) together with (3) and the analytical value of *He*_{0}^{(1,2)} (*a*, δ^{(1,2)}, 0), which is derived in section 2, we obtain

where