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

Authors


Abstract

[1] 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.

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

equation image

where Zν(t) is one of the Bessel or Hankel functions. Series expansions were developed by Dvorak and Kuester [1990] for the computation of Je0 (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 Je0 (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 Ye0 (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

equation image

where the lower limit of integration, δ, is chosen so that the integral converges at infinity in the complex t plane so that Zeν (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,

equation image

1.2. Current Work

[5] Theoretically, ILHIs of the Hankel type can be obtained via Je0 (a, ς) and Ye0 (a, ς), that is,

equation image

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

equation image
equation image

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

equation image

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:

equation image

where

equation image

and Zn = Jn, Yn, Hn(1), or Hn(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

equation image

[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 ∣aς∣. 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)]:

equation image
equation image

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

equation image
equation image

Furthermore, by employing (13) and (14) together with (3) and the analytical value of He0(1,2) (a, δ(1,2), 0), which is derived in section 2, we obtain

equation image
equation image

where

equation image

2. Complementary Incomplete Lipschitz-Hankel Integrals

[10] In order to relate the ILHIs to the CILHIs, we must find an analytical representation for the semi-infinite integral Zeν (a, δ, 0) in (3). For the case Re(a) ≥ 0, (6.611.6) and (6.611.7) in work by Gradshteyn and Ryzhik [1980] can be used to show that

equation image

where the principal branch is chosen for the natural logarithm, and the branch cut for equation image is

equation image

[11] For the case Re(a) < 0, the change of variables τ = t exp (∓jπ) can be applied to (2). This once again allows for the application of (6.611.6) and (6.611.7) from Gradshteyn and Ryzhik [1980], which yields

equation image

where Re (equation image) > 0 and the principal branch of the natural logarithm is again employed. Note that since (−a + equation image)(a + equation image) = 1 and ln (a + equation image)−1 = −ln (a + equation image), (20) is actually equal to (18). Thus (18) can be employed for all complex-valued a, where the branch cut for equation image is defined as in (17), and is shown in Figure 1. Substitution of (18) into (3) then allows the regular ILHI He0(1,2) (a, ς) to be related to the CILHI He0(1,2) (a, δ(1,2), ς).

Figure 1.

Branch cut for equation image shown in the complex a plane.

3. Problems With Round-Off Errors

[12] Now that we have introduced the CILHIs, we are in a position to investigate the cause of round-off errors. When ∣ς∣ is large, substitution of the first term of Hankel's asymptotic expansion [Watson, 1944, p.198, equations (5) and (6)] into (2) yields an approximation in terms of an incomplete Gamma function [Abramowitz and Stegun, 1972, equation (6.5.3)],

equation image

A simpler approximation can then be obtained by employing the first term in the asymptotic expansion for the incomplete Gamma function [(6.5.32), Abramowitz and Stegun, 1972],

equation image

Combining the results in (3), (18), and (22), we obtain the desired approximations for the ILHIs of the Hankel type

equation image

Approximations for the ILHIs of Bessel and Neumann types can now be obtained via (5) and (6), that is,

equation image
equation image

[13] By referring to the above equations, it can be seen that there are problems associated with round-off errors when using (4) to compute He0(1,2) (a, ς) when ∣Im(ς)∣ is large. For example, if we assume that Im(ς) ≫ 1, then ∣Je0 (a, ς)∣ and ∣Ye0 (a, ς)∣ will be relatively large because of the exponential growth exhibited by the sine and cosine functions in (24) and (25). The difference (i.e., see (4)) between the numerical results obtained from (24) and (25) will yield an accurate representation for He0(2) (a, ς) since the magnitude of this function is also relatively large. However, round-off errors will lead to inaccurate results if (4) is used to compute He0(1) (a, ς) since this result is very small according to (23). These errors arise in the case of He0(1) (a, ς) because two relatively large complex numbers are being added together to yield a relatively small number.

[14] This demonstrates that it is preferable to compute the ILHIs of the Hankel type, and then employ (5) and (6) to compute the ILHIs of the Bessel and Neumann types. Series expansions for the ILHIs and CILHIs of Hankel types are derived in sections 48.

4. First Convergent Factorial-Neumann Series Expansion

[15] 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

equation image

and the contribution due to the lower limit of integration, that is,

equation image

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. It can be shown that the first term in (27) does not have a square root branch cut in the complex a plane, that is, choosing either a plus or minus sign in front of equation image will yield the same result. Therefore we can choose the most convenient branch cut definition for equation image without affecting the form of (27).

[16] The expression in (27) is valid for all complex-valued a and ς provided that ∣a2 + 1∣ ≤ 1 and a ≠ 0. The region of convergence for this expansion is indicated in Figure 2 by the dark gray region.

Figure 2.

Regions of convergence for the first and second convergent factorial-Neumann series. The dark and light shaded regions show where the first and second convergent factorial-Neumann series, respectively, converge.

[17] When a = ±j, the k = 0 term is the only term left in the series expression for He0(1,2) (a, *, ς), and the natural logarithm term in (27) simplifies to ∓2j/(πa). Thus (27) can be expressed as

equation image

[18] Next, in order to find the first convergent factorial-Neumann series representation for the associated CILHI, the natural logarithm in (27) is expressed in a different form. Using equation (4.1.9) of Abramowitz and Stegun [1972],

equation image

it can be shown that

equation image

where the constants f(1,2) take on the values shown in Table 1. The expressions for the CILHIs are now obtained by combining the results in (3), (18), (27), and (30); that is,

equation image

where the constants C(1,2) are defined in Table 1 for different values of a, and the branch cut for equation image is defined as in (17) and shown in Figure 1. The convergent series representation for the indefinite integral He0(1,2) (a, *, ς) is given in (26). Since He0(1) (a, δ(1), ς) is singular at a = j and He0(2) (a, δ(2), ς) is singular at a = −j, only He0(1,2) (a, ς) can be computed for these cases.

Table 1. Variable Definitions for the Four Quadrants
 Quadrant IQuadrant IIQuadrant IIIQuadrant IV
f(1)−1−111
f(2)11−1−1
C(1) = −1 + f(1)−2−200
C(2) = −1 + f(2)00−2−2

5. Second Convergent Factorial-Neumann Series Expansion

[19] Since the first convergent factorial-Neumann series expansion has a limited region of convergence in the complex a plane (see Figure 2), we have found that it is useful to develop an additional convergent series expansion. The first step in the derivation of the new series expansion entails making the change of variables τ = sjat in (1); that is,

equation image

where

equation image

is defined to ensure that −js/a remains in the right half of the complex a plane. The sign function in the above equation is defined by

equation image

The multiplication theorem [Watson, 1944, p.142, equation (15)] can then be employed to rewrite (32) as

equation image

Note that we have ensured that Re(−js/a) ≥ 0, which is required to properly apply equation (15) of Watson [1944]. After applying the integral identities in (11.3.9) and (11.3.11) from Abramowitz and Stegun [1972], together with the series identity in (1.513) of Gradshteyn and Ryzhik [1980], (35) can then be rewritten in the form of (27), where the contribution to the indefinite integral is now expressed as

equation image

[20] The region of convergence for this series expansion is indicated in Figure 2 by the light gray region. We can see that the applicable region for the second convergent factorial-Neumann series is very different than for the first convergent factorial-Neumann series. Since the lower limit contribution is the same for both of the convergent factorial-Neumann series expansions, the CILHI representation for the second expansion can be obtained by substituting (36) into (31). For the special cases a = ±j, He0(1,2)j, *, ς) for the first and second convergent series yield the same results. Thus the resultant ILHI He0(1,2)j, ς) has the same form as in (28), as expected.

6. Struve Function Expansion

[21] Since the first convergent factorial-Neumann series expansion is not valid for a = 0 (see (26)), and this series converges slowly when a approaches zero, an additional series that is efficient for very small ∣a∣ is needed. The derivation of this new series is started by carrying out a Taylor series expansion for eat in (1), that is,

equation image

which is valid when ∣aς∣ is very small. The error introduced by this expansion is shown in Figure 3. Substituting (37) into (1) yields

equation image
Figure 3.

Relative error associated with replacing the exponential function with a two-term Taylor series expansion.

[22] The first integral on the left-hand side of (38) can be carried out analytically by employing (11.1.7) in [Abramowitz and Stegun, 1972], thereby yielding

equation image

where Hn(ς) are Struve functions of order n that can be computed by summing over Bessel functions of integer order, as shown in equations (12.1.19) and (12.1.20) of Abramowitz and Stegun [1972]. Meanwhile, after applying (5.56.2) of Gradshteyn and Ryzhik [1980] and (9.1.9) of Abramowitz and Stegun [1972], the second integral in (38) can be expressed as

equation image

Thus substitution of (39) and (40) into (38) yields the desired Struve function expansion

equation image

The CILHIs can then be obtained by using (3) and (18).

[23] Using (9.1.8) and (9.1.9) together with (12.1.4) and (12.1.5) of Abramowitz and Stegun [1972], the limitation of (41) as ς approaches zero can be easily shown to yield He0(1,2) (a, 0) = 0, which is consistent with (1).

7. Asymptotic Factorial-Neumann Series Expansion

[24] An asymptotic factorial-Neumann series expansion for the ILHIs of the Hankel type can be obtained by combining the results in equation (56) of Mechaik and Dvorak [1995] with those in equation (39) of Mechaik and Dvorak [1996]. However, recent tests have indicated that the branch cut for equation image that was chosen for the asymptotic factorial-Neumann series representations in these papers is not optimal for all values of the complex variables a and ς. Therefore in this paper we provide a brief overview of the analysis that was developed by Mechaik and Dvorak [1995, 1996], and use these results to find the optimal branch cut for all complex a and ς.

[25] Both asymptotic and convergent series expansions for the ILHIs of the Hankel type can be obtained by evaluating the following inverse Laplace transform [Mechaik and Dvorak, 1996]:

equation image

where Cu denotes the Bromwich contour, as shown in Figure 4. After closing the integration contour at ∣u∣ = ∞ in the left half of the complex u plane, residue theory can be used to show that

equation image

where

equation image
equation image

and the integration contours C± are shown in Figure 4. Note that the two terms in the square bracket in (42) end up canceling when the integrals along the contours on the two sides of the branch cuts emanating from u = −j and u = +j are added together in the cases of He0(1) and He0(2), respectively. Furthermore, it should also be noted that the branch cut for equation image in (44) must be chosen to be the same as the branch cut that is employed for equation image in (45).

Figure 4.

Inversion contour for the asymptotic factorial-Neumann series expansion in the complex u plane and the branch cut for equation image.

[26] In order to obtain an asymptotic expansion for the integral in (45), the pole term is expanded in a power series; that is, see equation (33) of Mechaik and Dvorak [1996]:

equation image

Substitution of this power series into (45) allows the integral to be integrated term by term [Watson, 1944, p.166, equations (4) and (5)], thereby yielding the desired asymptotic series representation

equation image

[27] While the power series representation in (46) converges rapidly in the vicinity of the branch points, that is, u = ±j, this series diverges for values of ∣u2 + 1∣ > ∣a2 + 1∣ that occur along the integration contours C±. Fortunately, because of the exponential decay factor in the integrand of (45), the series representation in (47) can be used as an asymptotic series when ∣ς(a2 + 1)∣ is large. In order to provide the most accurate asymptotic approximation to the integral in (45), the branch cut for the square root in (45) should be defined to maximize the rate of exponential decay in the integrand as u moves along the integration contour away from the associated branch point. This will minimize the error caused by the series in (46), which diverges when the integration variable u moves far enough away from the branch point. If we assume that the branch cuts associated with the branch points at u = ±j are defined at the angles ψ± relative to the positive real u axis (as shown in Figure 4), then these branch cuts are defined in the complex u plane by

equation image

where 0 ≤ τ < ∞. Since

equation image

along the contours C±, setting ψ± = π − ψ will provide the maximum rate of decay as τ increases along each of the branch cuts, where ψ = arg(ς). When computing He0(1), we only need to integrate along the contour C+. Therefore the branch cut associated with the branch point at u = +j is defined to lie at the angle ψ+ = π − ψ relative to the Re(u) axis. The other branch cut is left in the position ψ = π. Likewise, when He0(2) is being calculated, the angles for the two branch cuts are defined as ψ+ = π and ψ = π − ψ. With these choices for the branch cuts, the square roots for both equation image and equation image should be defined as

equation image

Thus the definition for the branch cut is dependent on the argument of ς. Mechaik and Dvorak [1995, 1996] fixed ψ± to π for all complex-valued ς, so the asymptotic series lost accuracy for complex-valued ς.

[28] In Figure 5, we plot the magnitudes of the terms in the series (47) for He0(1) (a, ς). It clearly illustrates that the series is an asymptotic series. The terms initially drop in magnitude very quickly, and then reach a local minimum near a point that we will refer to as kmax. For k > kmax, the terms then begin to increase in magnitude. The plot of the terms in the series (47) for He0(2) (a, ς) has a similar form, so because of space limitations, we will not plot the behavior of these terms in this paper. In order to obtain an accurate approximation for He0(1,2) (a, ς), this asymptotic series should be truncated when the magnitude of a term falls below the user defined error criteria. However, in order for this asymptotic series be useful, it must be truncated before reaching the first local minimum kmax. Otherwise, the series will diverge and yield an inaccurate result. A more detailed discussion about truncating the asymptotic series is included in section 9.

Figure 5.

Magnitude of the kth term in the asymptotic factorial-Neumann series of He0(1) (a, ς) for a = −2 + j0.35 and ς = j8.

[29] In order to find the asymptotic factorial-Neumann series representation for the associated CILHI that is consistent with the other series expansions, we rewrite (44) in terms of square roots whose branch cuts are defined as in Figure 1. Since the branch cuts in (50) and Figure 1 yield results that differ by a minus sign when Re(a) < 0 for the cases −π/2 < ±arg (±ja) < ∓ψ and ∓Im(a) > 1, and Re(a) = 0 and Im(a) < −1, we find that

equation image

For all other values of the parameter a, the two branch cut definitions yield results that are equivalent, so we can directly employ the expression in (44). The desired CILHI representation is now obtained by adding the results in (18) and (43) (see (3)), that is,

equation image

where

equation image

and the contribution due to the integration along the branch cut is given in (47).

8. Quasi-Neumann Series Expansion

[30] The quasi-Neumann series expansion can be obtained by employing the results of Mechaik and Dvorak [1996, equation (53)], that is,

equation image

where γ is Euler's constant (γ = 0.577216…) and E1(·) is the exponential integral function given by equation (5.1.1) of Abramowitz and Stegun [1972]. Note that the expression by Mechaik and Dvorak [1996] for the quasi-Neumann series expansion contains typographical errors. The branch cut associated with the square root in (54) is shown in Figure 6, and defined mathematically as

equation image
Figure 6.

Branch cut for equation image shown in the complex a plane.

[31] The functions Jek(a, ς; m) that appear in (54) are related to the regular ILHIs, and are defined as

equation image

The terms Jek(a, ς; 0), which are required to evaluate the sum in (54), are found by applying the technique given by Olver [1967] and Olver and Sookne [1972] to the recurrence relation [Mechaik and Dvorak, 1996]

equation image

with the normalization condition

equation image

While the series expansion in (54) converges for all complex-valued a and ς, the expansion is most useful when ∣ς∣ is small to moderate.

[32] The series expansions for the associated CILHIs can be obtained by applying (3) and (18). Unfortunately, the form of the quasi-Neumann series expansion prohibits the analytical cancellation of the terms in (18) with those in (54). This results in numerical instabilities for some values of a and ς. Since the other CILHI series representations do not suffer from this difficulty, the quasi-Neumann series expansion should be used as the method of last resort to compute the CILHIs of the Hankel type.

9. Numerical Computation of He0(1,2) (a, ς) and He0(1,2) (a, δ(1,2), ς)

[33] The series expansions that were presented in the previous sections have very different properties for different values of the input parameters a and ς. In order to illustrate this, in Table 2 we list the improvements in the computational efficiency and the achieved relative errors for the different series expansions for fixed input parameters a = 0.2 + j0.5 and ς = 12 + j10. The improvement in the efficiency is measured by taking the ratio of the amount of CPU time required to evaluate (1) by numerical integration to the time required to evaluate He0(1,2) (a, ς) by the various series expansions. An adaptive quadrature algorithm that is based on the Guass 30-point and Kronrod 61-point rules is employed for the numerical integration. The relative error was obtained by comparing the numerical values obtained by using the series expansions with the values obtained by using the numerical integration algorithm, where the numerical integration values were assumed to be exact. The following key is used: 1st conv. represents the first convergent factorial-Neumann series expansion, which is computed by using (27), where He0(1,2) (a, *, ς) is defined in (26), (when a = ±j, use (28) instead); 2nd conv. means the second convergent factorial-Neumann series, which is also computed by using (27), but He0(1,2) (a, *, ς) is defined in (36); asymptotic refers to the asymptotic factorial-Neumann series, which is obtained by equations (43) and (47), Neumann stands for the quasi-Neumann series expansion, which is defined in (54), and Struve is the Struve function series expansion, which is defined in equation (41).

Table 2. Efficiency and Accuracy of the Various Series Expansions for Fixed a = 0.2 + j0.5 and ς = 12 + j10a
 He0(1) (a, ς)He0(2) (a, ς)
Efficiency ImprovementRelative ErrorbEfficiency ImprovementRelative Error
  • a

    Here 1st conv. represents the first convergent factorial-Neumann series expansion, and 2nd conv. means the second convergent factorial-Neumann series. N/A means the related series is not applicable.

  • b

    Read 1.65E-05 as 1.65 × 10−5.

1st conv.261.65E-05362.04E-06
2nd conv.N/AN/AN/AN/A
Asymptotic274.61E-07421.33E-03
StruveN/AN/AN/AN/A
Neumann235.75E-05233.32E-07

[34] From Table 2, we see that while more than one series expansion can be used to compute He0(1,2) (a, ς) for a given combination of a and ς, the accuracies and efficiencies of these series expansions are different. Therefore in this section we present a procedure for selecting the proper expansion to use for the accurate numerical computation of the indefinite integral He0(1,2) (a, *, ς) (see (26), (36), and (47)) on the basis of the values of the input variables a and ς. Once this portion of the integral has been computed, then He0(1,2) (a, ς) and He0(1,2) (a, δ(1,2), ς) are obtained by adding on the lower-limit contributions, that is, (27), (31), (43), and (52). Note that the Struve function expansion and quasi-Neumann series expansion are the exceptions. For these two expansions, the regular ILHIs He0(1,2) (a, ς) are first computed, and then the CILHIs are obtained by applying (3) and (18).

[35] For the special cases of a = ±j, (28) and (31) are used to compute He0(1,2)j, ς) and He0(1,2)j, δ(1,2), ς) respectively, for all ς, since they are the simplest and thus most efficient expressions. In addition, (28) and (31) remove the apparent singularities at a = ±j.

[36] Next, when ∣ς∣ < SD + 5 and ∣aς∣ < ɛ, the Struve function expansion (41) is chosen as the first priority. Here ɛ is the corresponding value of ∣aς∣ in Figure 3 for the given relative error 10SD, and SD denotes the user specified number of significant digits for the results.

[37] Otherwise, when ∣ς∣ > SD + 5, and ∣ς(a2 + 1)∣ ≫ 1, it is most efficient to use the asymptotic factorial-Neumann series if possible. However, since it is an asymptotic series, and does not converge, we must determine when to truncate the series. If we wish to employ the asymptotic factorial-Neumann series expansion (47) to compute He0(1,2) (a, *, ς) to SD significant digits of accuracy, then the following condition must be met

equation image

where we have assumed that k and ∣ς∣ are large enough that we can employ Hankel's asymptotic expansion for the Hankel functions [Watson, 1944, p.198, equations (5) and (6)] and Stirling's formula for the Gamma function of Abramowitz and Stegun [1972, equation (6.1.37)]. The estimation of ∣He0(1,2) (a, *, ς)∣ can be obtained by combining (22) with (52),

equation image

Substitution of (60) into (59) then yields

equation image

This result can then be further simplified as

equation image

for cases when A(1,2) = 0 (see (53)).

[38] In order to determine whether the asymptotic factorial-Neumann series expansion can be employed, we must find the value of k that minimizes the right-hand side of (61), that is, minimizes ∣(2k + 1)/[eς(a2 + 1)]∣k, and then see if the inequality in (61) is satisfied. As demonstrated by Dvorak and Kuester [1990], the local minimum is achieved when

equation image

Since we have assumed that ∣ς∣ > k in the derivation of (59), the summation in the asymptotic factorial-Neumann series expansion (47) should not be carried out past

equation image

since the asymptotic series diverges for k > kmax. Thus the asymptotic factorial-Neumann series expansion can be employed if (61) is satisfied for k = kmax.

[39] If the asymptotic factorial-Neumann series expansion will not provide the desired accuracy for a given set of inputs, then one may be able to employ one of the two convergent factorial-Neumann series expansions. Recall that the first and second convergent factorial-Neumann series expansions (i.e., (26) and (36)) converge for all complex-valued a, provided ∣a2 + 1∣ ≤ 1, a ≠ 0 and ∣a2 + 1∣ ≤ ∣a2∣, respectively. However, reference to Figure 2 shows that there is a region of overlap where both of these expansions converge. Since the first convergent factorial-Neumann series expansion involves a summation over Hankel functions whose arguments are the same as those in the other series, we employ (26) whenever possible. Numerical tests have shown that this expansion can be employed when ∣a2 + 1∣ ≤ 0.87. If this condition is not satisfied, then the second convergent factorial-Neumann series expansion (36) is utilized if ∣a2 + 1∣ ≤ 0.87∣a2∣. The regions in the a plane where we have chosen to employ these two convergent series expansions are shown in Figure 7. A comparison between Figures 2 and 7 shows that a reduced coverage of the a plane is the price that must be paid for improved accuracy and efficiency.

Figure 7.

Regions in the complex a plane where the two factorial-Neumann series expansions are employed. The dark and light shaded regions show where the first and second convergent factorial-Neumann series, respectively, are employed.

[40] Finally, if none of the other series can be used, then the quasi-Neumann series expansion (54) is utilized to compute the ILHIs for all remaining values of a and ς.

[41] The required sequence of Hankel functions can be computed by the algorithm outlined in the next section, and the set of Bessel functions can be calculated by applying the algorithm outlined by Dvorak and Kuester [1990].

10. Computation of the Required Hankel Functions

[42] The two convergent factorial-Neumann series (26) and (36), and the asymptotic factorial-Neumann series (47), contain sums over Hankel functions of integer order. Heckmann and Dvorak [2001] showed that Hk(1)(ς) and Hk(2)(ς) behave differently with respect to increases in order for different values of Im(ς). The magnitude of Hk(1)(ς) is monotonically increasing as the order increases from k = 0,1,2…, when Im(ς) > 0. On the other hand, ∣Hk(1)(ς)∣ passes through a local minimum at some order k = r when Im(ς) < 0. Conversely, the magnitude of Hk(2)(ς) has the opposite behavior with respect to Im(ς); that is, the sequence possesses a local minimum when Im(ς) > 0 and is monotonically increasing with respect to order when Im(ς) < 0. On the basis of this behavior, Heckmann and Dvorak [2001] outlined two different algorithms to compute the set of Hankel functions. For the monotonically increasing case, the recurrence relation [Abramowitz and Stegun, 1972, equation (9.1.27)]

equation image

is employed in the forward direction to compute the higher-order Hankel functions after obtaining the zero- and first-order starting functions. For the nonmonotonic case, a hybrid forward/backward recurrence algorithm is used. For example, starting from the order where the local minimum occurs, (65) is employed in the backward and forward directions to compute the lower-order and higher-order Hankel functions, respectively. However, we have found that obtaining an accurate local minimum for the Hankel functions, as described in section 6 of [Heckmann and Dvorak, 2001], requires that the starting values H0(1,2)(ς) and H1(1,2)(ς) possess a large number of significant digits of accuracy, sometimes even exceeding the maximum number of significant digits that the algorithms for computing H0(1,2)(ς) and H1(1,2)(ς) can provide. This accuracy requirement either requires more time to compute H0(1,2)(ς) and H1(1,2)(ς) for cases where the algorithms can satisfy the significant digits requirement, or else an accurate local minimum cannot be obtained so the other Hankel functions cannot be accurately computed.

[43] In order to overcome this problem, we have developed a new algorithm for the accurate and efficient computation of Hk(1,2)(ς) for all complex-valued ς. This new algorithm does not require finding the location of the local minimum, so less accuracy is required for the two starting values H0(1,2)(ς) and H1(1,2)(ς). Furthermore, the new algorithm still yields the same level of accuracy for higher orders of Hk(1,2)(ς). A detailed discussion about the new algorithm will be covered in a separate paper. However, for convenience, the algorithms are summarized below.

[44] When Im(ς) > 0, the ∣Hk(1)(ς)∣ are monotonically increasing for all k. Therefore (65) can be applied safely in the forward direction. Thus the starting functions, H0(1)(ς) and H1(1)(ς), are first computed by the algorithm described in section 4 of [Heckmann and Dvorak, 2001]. Next, the higher-order Hankel functions are computed by forward recurrence using (65). Once an accurate sequence of Hankel functions is obtained, that is, Hk(1)(ς), then we can calculate Hk(2)(ς) by using

equation image

where it has been shown by du Toit [1990] that Jk (ς) is a monotonically decreasing function with respect to order for complex-valued ς. Therefore the sequence Jk (ς) can be calculated using backward recurrence to a user defined number of significant digits, as outlined by Dvorak and Kuester [1990]. When Im(ς) < 0, the behaviors for the Hankel functions of the first and second kinds interchange, as do the procedures for computing the higher-order Hankel functions. For purely real ς, either of the above two algorithms can be applied since the real ς case can be treated as a limit where Im(ς) → 0. However, if high accuracy is required for both the real and imaginary parts of Hk(1,2)(ς), then

equation image

must be used, where Jk (ς) is computed using the algorithm outlined by Dvorak and Kuester [1990] and Yk (ς) is calculated by the algorithm given by du Toit [1990].

[45] When ∣ς∣ is large, and only a few orders of Hk(1,2)(ς), k = 0,1,2,.m are needed, where m is much less than the order of the local minimum r, then (65) can be used in the forward direction for the nonmonotonic case without a serious loss of accuracy. This helps to expedite the calculation of the nonmonotonic Hankel functions, since it is no longer necessary to compute Jk (ς) by backward recurrence, which takes a long time when ∣ς∣ is large.

11. Comparisons With Numerical Integration

[46] In this section, the numerical results that are produced by numerical integration of (1) are used to verify the accuracy of the various series expansions. One of the difficulties of applying numerical integration involves the logarithmic singularity at t = 0 that is encountered during the computation of He0(1,2) (a, ς) (see (1)). In order to avoid this singularity, we have found that it is better to numerically integrate He1(1,2) (a, ς), which has a removable singularity. He0(1,2) (a, ς) is then obtained via backward recurrence using (8).

[47] Tables 3 and 4 provide examples showing how He0(1) (a, ς) and He0(2) (a, ς) are computed for various combinations of the parameters a and ς. The rules that are discussed in section 9 were implemented to choose the optimal expansion for evaluating He0(1,2) (a, ς) for each set of parameters. Five significant digits (SD = 5) of accuracy were requested for all the calculations in Tables 3 and 4. The first and second columns in Tables 3 and 4 list typical values of the parameters a and ς that we tested. The third column shows which series was used to evaluate He0(1,2) (a, ς) for given values of a and ς. The keys that denote the various expansions are the same as those used in Table 2.

Table 3. Computational Results for He0(1) (a, ς)
aςSeriesaEfficiency ImprovementRelative Errorb
  • a

    Here 1st conv. represents the first convergent factorial-Neumann series expansion, and 2nd conv. means the second convergent factorial-Neumann series.

  • b

    Read 3.13E-12 as 3.13 × 10−12.

0 + j05 + j2Struve433.13E-12
− 0.03 + j0.010.01 + j0.01Struve462.93E-08
0.2 + j0.352 + j5Neumann254.28E-07
2 − j0.356 + j0Neumann223.12E-07
−2 + j0.350 + j8Asymptotic428.02E-09
0 + j2 + j61st conv.457.41E-12
− 0.2 + j0.92 + j161st conv.1011.77E-06
0.2 − j0.916 − j21st conv.331.23E-07
0 − j0 + j151st conv.400
0.3 + j1.50 + j62nd conv.182.91E-06
0 − j210 + j02nd conv.151.37E-06
0 + j00 + j36Asymptotic4231.57E-10
0.2 + j0.3510 + j36Asymptotic4162.93E-10
0.3 − j1.536 + j0Asymptotic3473.23E-10
2 + j1.510 − j36Asymptotic8121.09E-10
Table 4. Computational Results for He0(2) (a, ς)
aςSeriesaEfficiency ImprovementRelative Errorb
  • a

    Here 1st conv. represents the first convergent factorial-Neumann series expansion, and 2nd conv. means the second convergent factorial-Neumann series.

  • b

    Read 1.76E-12 as 1.76 × 10−12.

0 + j05 + j2Struve441.76E-12
−0.03 + j0.010.01 + j0.01Struve452.99E-08
0.2 + j0.352 + j5Neumann267.75E-08
2 − j0.356 + j0Neumann232.81E-07
−2 + j0.350 + j8Neumann225.25E-09
0 + j2 + j61st conv.452.61E-15
− 0.2 + j0.92 + j161st conv.343.92E-08
0.2 − j0.916 − j21st conv.342.27E-07
0 − j0 + j151st conv.381.44E-16
0.3 + j1.50 + j62nd conv.212.98E-07
0 − j210 + j02nd conv.165.74E-06
0 + j00 + j36Asymptotic1433.70E-07
0.2 + j0.3510 + j36Asymptotic1434.85E-07
0.3 − j1.536 + j0Asymptotic3531.09E-10
2 + j1.510 − j36Asymptotic8111.56E-10

[48] The fourth column in Tables 3 and 4 lists the improvement in the efficiency that resulted from using the series expansions as compared to the numerical integration algorithm. The efficiency is defined the same way as for Table 2. As shown in Tables 3 and 4, it takes the numerical integration algorithm much longer to evaluate He0(1,2) (a, ς) than the chosen series expansion. When ∣ς∣ is large, the improvement in efficiency is tremendous. The reason for this is that as ∣ς∣ gets larger, the integrand of (1) also becomes more oscillatory, so it takes more time to compute the integral using numerical integration. On the other hand, when ∣ς∣ is large, we can use the asymptotic factorial-Neumann series, which only needs a few terms. Therefore it takes much less time.

[49] The fifth column in Tables 3 and 4 displays the relative error in the results, where the relative error is defined the same way as in Table 2. The small relative errors that are shown in Tables 3 and 4 indicate that the series expansions can be used to accurately compute He0(1,2) (a, ς). The algorithm for computing He0(1,2) (a, ς) was tested over a wide range of input values for a and ς, and we found that good accuracy can be achieved by using the optimal series in all the cases we tested.

12. Characteristics of the ILHIs and CILHIs

[50] In the previous sections, we first defined the ILHIs and CILHIs of the Hankel type. We then developed various series expansions for computing these special functions. Numerical comparisons with a numerical integration algorithm showed that these series are accurate and efficient. Now, in this section, we explore the behavior of the ILHIs and CILHIs of the Hankel type in more depth. This will help to provide a better understanding for this group of special functions.

[51] Using (1), we can easily show that He0(1,2) (a, ς) does not have a branch cut in the complex a plane. However, it does have a logarithmic branch cut along the negative ς axis in the ς plane, which is caused by H0(1,2)(ς). The branch point for He0(1,2) (a, ς), which is located at ς = 0, is not a singular point as in H0(1,2)(ς), but a zero point. The magnitudes of He0(1) (a, ς) and He0(2) (a, ς) are plotted in Figures 8a and 8b, respectively, for the special case a = j. From these plots we can clearly see the branch cut in the complex ς plane that extends along the negative real ς – axis.

Figure 8.

(a) Three-dimensional view of ∣He0(1) (j, ς)∣ plotted in the complex ς plane. (b) Three-dimensional view of ∣He0(2) (j, ς)∣ plotted in the complex ς plane.

[52] For He0(1,2) (a, δ(1,2), ς), the branch cut is a little more complicated. From (3), we see that He0(1,2) (a, δ(1,2), ς) not only has a ς plane branch cut like He0(1,2) (a, ς), but it also has a branch cut in the complex a plane that is associated with He0(1,2) (a, δ(1,2), 0). In order to help illustrate the location of the branch cut in the complex a plane, Figures 9a and 9b show 3-D views of the imaginary parts of He0(1) (a, δ(1), 0) and He0(2) (a, δ(2), 0), respectively. The imaginary parts of He0(1,2) (a, δ(1,2), 0) are plotted since these plots most clearly show the branch cut.

Figure 9.

(a) Three-dimensional view of the imaginary part of He0(1) (a, δ(1), 0) plotted in the complex a plane. (b) Three-dimensional view of the imaginary part of He0(2) (a, δ(2), 0) plotted in the complex a plane.

[53] From Figure 9a, we see that He0(1) (a, δ(1), 0) has a branch cut in the complex a plane, which emanates from a = j to a = j∞ along the positive imaginary axis. Furthermore, the branch point at a = j is a singular point. Likewise, the branch cut in the a plane for He0(2) (a, δ(2), 0) emanates from a = −j to a = −j∞ along the negative imaginary axis, where a = −j is a singular point. These branch cuts in the a plane can also be obtained by investigating (18), and are caused by the square root terms in He0(1,2) (a, δ(1,2), 0). When ∣a∣ approaches infinity, it is easy to show that He0(1,2) (a, δ(1,2), 0) → 0. This means that

equation image

[54] Another important characteristic of ILHIs and CILHIs of the Hankel type is their behavior when ∣ς∣ is large. We will let a = aR + jaI and ς = ςR + jςI, where aR, aI, ςR and ςI are real variables. Substituting these expressions into (22) yields

equation image

It is clear that the magnitude of He0(1,2) (a, δ(1,2), ς) is dominated by the exponential term exp [−aRςR + (aI ∓ 1)ςI] if −aRςR + (aI ∓ 1)ςI ≠ 0 and ∣ς∣ is large. Otherwise, when −aRςR + (aI ∓ 1)ςI = 0, He0(1,2) (a, δ(1,2), ς) only decays as the order 1/equation image. In addition, if we look at the second exponential term in (69), we see that with increases of ∣a∣ and ∣ς∣, He0(1,2) (a, δ(1,2), ς) becomes more oscillatory.

[55] Combining (69) and (18), we find the expression for He0(1,2) (a, ς), that is,

equation image

In (70), the term −He0(1,2) (a, δ(1,2), 0), which is only a function of a, is added to the asymptotic expression for He0(1,2) (a, δ(1,2), ς). This means that for a fixed value of a, although He0(1,2) (a, δ(1,2), ς) converges to zero as ∣ς∣ increases, He0(1,2) (a, ς) converges to a constant, which may be nonzero.

[56] Next we use an example to illustrate the different behaviors of He0(2) (a, ς) and He0(2) (a, δ(2), ς), and provide some insight into the role that He0(2) (a, δ(2), ς) plays in the formulation of problems. In work by Heckmann and Dvorak [1999], ILHIs appeared in a combination with trigonometric functions, that is,

equation image

where a is a purely imaginary number between −j and j, ς is a complex-valued variable with a very small positive real part, and Im(ς) ≤ 0. In Figure 10a, ∣eaςHe0(2) (a, ς)∣ is plotted on a logarithmic scale. This plot shows that one of the first two terms on the right-hand side of (71) increases exponentially with increasing ∣Im(ς)∣. In addition, the third term in (71) also exhibits an exponential increase with ∣Im(ς)∣. This may cause a loss of accuracy when ∣Im(ς)∣ is moderate to large if the large exponential terms are not removed analytically. Note that the exponentially increasing behavior in the first two terms of (71) is caused by the fact that ∣He0(2) (a, ς)∣ approaches a nonzero value, that is, ∣He0(2) (a, δ(2), 0)∣, when ∣Im(ς)∣ is large. If we can remove the nonzero value from He0(2) (a, ς), we might be able to remove the large exponential terms from the expression for Ω. After substituting (3) into (71), and combining the results in (18) and (30), Ω can then be expressed in terms of CILHIs,

equation image

In this expression, the third term is always an exponentially decaying function, and the first two terms, as shown in Figure 10b, also no longer exhibit an exponentially increasing behavior. This demonstrates that the large exponential terms have been removed analytically by expressing Ω in terms of CILHIs. Thus (72) is better behaved numerically than (71). For the special cases of a = ±j, where He0(2) (a, δ(2), ς) is singular, the exponential terms can be removed by substituting (28) and (30) into (71).

Figure 10.

(a) ∣eaςHe0(2)(a, ς)∣ plotted as a function of a and ς, where Re(ς) is fixed to 10−6 and Re(a) = 0. (b) ∣eaςHe0(2) (a, δ(2),ς)∣ plotted as a function of a and ς, where Re(ς) is fixed to 10−6 and Re(a) = 0.

13. Application in Electromagnetics That Involves Computation of He0(2) (a, δ(2), ς)

[57] In this section, we use an example from Z. Zhu et al. (Extension of an efficient moment method based, full-wave layered-interconnect simulator to finite-width expansion functions, submitted to IEEE Transactions on Advanced Packaging, 2004, hereinafter referred to as Zhu et al., submitted manuscript, 2004) to demonstrate the improvement in efficiency that results from using the CILHI expansions for an engineering application. The application involves determining the current that flows on a stripline circuit. Figure 11 shows a short section of interconnect that resides between two ground planes. The fields were first formulated in the spectral domain and then an electric field integral equation was formulated for the current on the interconnect. The method of moments (MOM) was then used to reduce the integral equation to a matrix equation, which was used to find the unknown current on the trace. In the MOM, the signal traces are subdivided into many cells, which are associated with rooftop expansion functions as shown in Figure 11. The testing functions are chosen to have the same form as the expansion functions to achieve better convergence in the solution for the current. The reaction between any two of these cells, that is, the reaction element, is then calculated to fill the reaction matrix. The reaction elements take the form of 2-D Sommerfeld integrals, which are highly oscillatory and slowly convergent. Therefore it is difficult to compute the required reactions directly by using numerical integration.

Figure 11.

Layout in the x-y plane of a thin interconnect that is embedded in a homogeneous dielectric and shielded by two perfect electric conductor planes. Only one rooftop expansion function is illustrated.

[58] Zhu et al. (submitted manuscript, 2004) showed that the reaction integrals can be carried out analytically in terms of CILHIs of the Hankel type. The advantage of the CILHI representation is that it is computationally more efficient than a numerical integration algorithm since the CILHIs can be computed by efficient series expansions. Figure 12 shows the improvement in the efficiency that is achieved by using the CILHIs as compared to using an adaptive quadrature numerical integration algorithm. The program was run on a PC with an Intel Pentium 4 CPU (3.0GHz) and 512M memory. Figure 12 clearly shows that employing the CILHIs can save a tremendous amount of computation time; that is, it is at least 2 orders of magnitude faster than numerical integration. The CPU times required to compute the MOM reaction elements using the CILHI representations were approximately the same as the spacing between the expansion and testing cells increases. The reason for this is that the optimal series expansion for the CILHIs is automatically chosen for each set of input parameters according to the rules discussed in section 9. On the other hand, the integrand of the MOM reaction element becomes more and more oscillatory as the cell separation increases, thus requiring many more points in order for the adaptive numerical integration algorithm to converge. Therefore the improvement in efficiency is even more significant for large separations; that is, a 4 order of magnitude improvement is seen when the expansion and testing cells are separated by 0.1λ.

Figure 12.

Computation times for a reaction element versus the spacing between the centers of the expansion and testing functions when using numerical integration and the closed-form representation that involves CILHIs of the Hankel type.

14. Conclusions

[59] In this paper, we have demonstrated how ILHIs and CILHIs of Hankel form can be accurately and efficiently computed for different values of the complex variables a and ς by using a combination of five series expansions. We have shown that direct computation of the ILHIs and CILHIs of Hankel form is necessary to avoid the round-off errors that are encountered when these functions are computed by adding ILHIs of the Bessel and Neumann types. If desired, the ILHIs of the first and second Hankel types can then be used to accurately compute ILHIs of the Bessel and Neumann forms. A new algorithm for computing Hankel function is also briefly discussed.

[60] The first convergent factorial-Neumann series and the quasi-Neumann series were taken directly from previous papers. However, the second convergent factorial-Neumann series and the Struve function expansion are new. Furthermore, we have also found the optimal branch cut for the asymptotic factorial-Neumann series expansion that improves its accuracy. Finally, we have shown that it is preferable to employ CILHIs in problems when ∣Im(ς)∣ is large, or more generally, when ∣Re(aς)∣ is large.

Acknowledgments

[61] This work was supported in part by the National Science Foundation's Center for Low-Power Electronics (CLPE) under grant EEC-9523338 and the Semiconductor Research Corporation (SRC) under contract 2001.NJ.956.001.

Ancillary