Exactly calculable field components of electric dipoles in planar boundary

The Sommerfeld integrals for the electromagnetic ﬁelds in the planar boundary between air and a homogeneous, isotropic medium, due to a horizontal and a vertical electric dipole each lying along the interface, are examined in detail. In the case of the horizontal dipole, the tangential electric ﬁeld is given in terms of series that involve conﬂuent hypergeometric functions, namely, the Fresnel and exponential integrals. A similar exposition is presented for the magnetic and vertical electric ﬁelds of the vertical dipole. When the index of refraction of the adjacent space is of a sufﬁciently large magnitude, the derived series converge rapidly and uniformly with the distance from the source. Speciﬁcally, their rates of convergence are shown to be independent of distance. It is pointed out that the corresponding formulas of King et al. are valid down to any distance close to the source, where they smoothly connect to known ‘‘quasi-static’’ approximations. © 2001 American Institute of Physics. @ DOI: 10.1063/1.1330731 #


I. INTRODUCTION
Almost a century ago, Sommerfeld 1 first formulated the problem of the radiating vertical electric dipole located in the planar boundary between two homogeneous and isotropic half spaces by invoking the Hertz vector ⌸ rather than the electromagnetic fields E and B. With the use of the Fourier-Bessel representations in cylindrical coordinates, Sommerfeld proposed approximate formulas for ⌸ for distances of many wavelengths in air away from the source.Soon after, his student Ho ¨rschelman 2 applied the same method to the case of the horizontal electric dipole in air.Other authors revisited these problems aiming at alternative representations for ⌸ that could be amenable to asymptotic evaluations for sufficiently large distances.A historical account and extensive list of references can be found in the monograph by Ban ˜os. 3 Serious efforts to derive accurate expressions for the ⌸ of electric or magnetic dipoles were often made under the simplifying yet practically significant assumption that both the source and the observation point lie at the interface. 3-10Some of the components of ⌸ then involve the Fourier-Bessel integrals, 3 where k j ( jϭ1, 2) is the complex wave number in medium j, is the polar distance from the source, and J 0 is the Bessel function of order 0. Van der Pol 6 showed that U() is given in terms of elementary functions, while V() can be converted to a finite, one-dimensional integral of an elementary function that readily yields Sommerfeld's approximate result.On the basis of Van der Pol's formula for V(), Rice 10 derived exact series expansions that, although credited as being ''uniform'' in the distance , 11 become impractical when ͉k j ͉ӷ1.He also proposed disparate asymptotic expansions for these distances when the refraction index k 1 /k 2 is close to 1, k 2 being the wave number in air.Along the same lines is the exposition by Wise. 7Noteworthy is Fock's 9 expansion for V() in terms of products of Bessel functions with half-integer indices, where the expansion parameter (k 1 2 Ϫk 2 2 )/(k 1 2 ϩk 2 2 ) is assumed to be of magnitude less than 1.Ban ˜os 3 rederived the Sommerfeld-Van der Pol formula by applying a version of the steepest descent method, where a simple pole is extracted from the vicinity of a saddle point, and neglecting high orders in k 2 /k 1 .However, the issue of connecting this formula, which is valid in the range k 2 3 /͉k 1 2 ͉уO(1), to the respective approximation for ͉k 1 ͉Ӷ1 was not essentially addressed.In a series of works, 12 Wait gave asymptotic formulas for the Sommerfeld integrals in different ranges of polar distances and source heights.Consider, for example, the ranges k 2 Ӷ1, k 2 ϭO(1), and k 2 ӷ1 when the dipole and the observation point both lie in the boundary; 13 even in this simplest nontrivial case, Wait's approximations seem to be based on intuitive arguments.In particular, in the ''quasi-static approach,'' 13 the fields in air are regarded as solutions of Laplace's or Poisson's equation with no practical restriction on k 1 , but there is no clear indication, for instance, about the convergence or the magnitude of the remainder of the underlying expansion when k 1 ϭO(k 2 ) with k 2 Ͻ͉k 1 ͉.In the spirit of the quasi-static approach, the computation of the Hertz vector is carried out in Refs.14-16 for low frequencies via a convenient resummation of the -Maclaurin expansion for the radical under the integral sign.The ensuing simple expressions are interpreted as superpositions of primary and reflected fields, where the earth is replaced by a perfectly conducting medium with the boundary being shifted by the distance 1/k 1 . 16Notably, the electric and magnetic fields are obtained through direct differentiations of the approximate formulas for ⌸.
Recently, integrated formulas were derived by King et al. 17 for the electromagnetic field in air over an imperfectly conducting or dielectric earth when the source is a horizontal or vertical electric dipole.Their major simplifying conditions are k 2 2 Ӷ͉k 1 2 ͉ and k 2 rϾO(1), r being the radial distance from the source.Some of the novelties of their approach can be outlined as follows.First, these authors deal directly and systematically with the field itself and not the Hertz vector; their set of formulas satisfy Maxwell's equations and the required boundary conditions consistently to the desired order in k 2 2 /k 1 2 .Second, in their sequence of approximation steps, the direct and the ideal-image fields are singled out, some of the remaining integrals are computed exactly by analytical means, and large-argument approximations for the Bessel functions are only applied to the remainders that involve the Sommerfeld pole.The results advance the works of Ban ˜os 3 and others both quantitatively, with the retainment of a larger number of terms, and qualitatively, with the notions of the surface and lateral waves being dissociated in the mathematical treatment from that of a saddle-point in the vicinity of a pole.
In a recent paper, 18 King and Wu make use of the approximate formulas of Ref. 17 for the horizontal dipole to calculate the electromagnetic field in air of infinitely long transmission lines above the earth.However, as pointed out in Ref. 18, the violation of the condition ͉k 1 r͉Ͼ1 at extremely low frequencies introduces an inaccuracy for the axial component of the electric field.A formula for this component that is uniform in distance was later derived in a more elaborate analysis by Margetis. 19The inaccuracy mentioned above signifies one of the instances where approximate formulas that are known to hold sufficiently far from the source are forced to be extended to distances too close to the source.An interesting question is whether it is possible, and if so in what sense, to connect the lateral-wave formulas of King et al. 17 to known near-field expressions, such as those given by Wait for k 2 Ӷ1, 13 so that the final formulas adequately describe the field for all reasonable distances when k 2 2 Ӷ͉k 1 2 ͉. 20 Various interesting references and formulas for the evaluation of Sommerfeld-type integrals are provided in Ref. 21. Noteworthy among these formulas are the representations in terms of incomplete cylindrical functions.
The purpose of this paper is twofold.The first is to evaluate exactly, in terms of series that are uniform in , those Sommerfeld integrals that are given by integrals of elementary functions, by relaxing the condition k 2 2 Ӷ͉k 1 2 ͉.This task is carried out in Secs.III and IV for the electromagnetic field of electric dipoles lying in the planar interface; the use of the Hertz vector is entirely avoided, in the spirit of Ref. 17.The expansion parameter is the inverse of the refraction index, k 2 /k 1 , which is assumed to be of magnitude less than 1, and the coefficients are known transcendental functions, namely, the exponential and Fresnel integrals.These series are believed to be new.In particular, the rates of convergence of the derived series are shown to depend only on the ratio k 2 2 /k 1 2 . 11Emphasis is also placed on obtaining bounds and estimates for the remainders when a finite number of terms are summed.As a consequence, stringent conditions for the validity of simplifications under k 2 2 Ӷ͉k 1 2 ͉ can follow.All derivations are subject to routine mathematical rigor, and comparisons with numerical computations are beyond the scope of this paper. 22A discussion on the merits of the present analysis for numerical evaluations is provided in Sec.VI.
The second purpose is to demonstrate that the corresponding lateral-wave formulas in Ref. 17 may indeed be extended to distances from the source that are short compared to the wavelength in air.In Sec.V we deal precisely with this task via the step-by-step approximations of the exact series.Finally, in Appendix A we calculate analytically a class of integrals involving Bessel functions through a generalized Schwinger-Feynman representation; Van der Pol's formula 6 essentially follows as a special case.The nature of the field asymptotic expansions for k 2 ӷ1 is analyzed in Appendix B on the basis of the derived series, while in Appendix C we revisit the simplifications of the original integrals in the limiting cases k 2 Ӷ1 and k 2 ӷ1.The time dependence e Ϫit is suppressed throughout the analysis.

A. Horizontal electric dipole
The geometry and Cartesian coordinate system are shown in Fig. 1.As the source and the observation point approach the boundary from below (d→0 ϩ ) and from above (z→0 ϩ ), respectively, the Fourier-Bessel representation for the electromagnetic field in the cylindrical coordinates (,,z) with xϭ cos and yϭ sin (0рϽ2) is 17 1.The geometry and Cartesian coordinate system for a unit horizontal dipole in the earth.The height d is allowed to approach zero (d→0 ϩ ).
the first subscript in each component referring to the region ͑1 for zϽ0 and 2 for zϾ0).These integrals are divergent in the conventional sense.The procedure implied by allowing d→0 ϩ and z→0 ϩ in Fig. 1 dictates that they be interpreted in the sense of Abel. 23The first Riemann sheet is such that ( jϭ1, 2) with the branch-cut configuration of Fig. 2 where k 1 is taken to be real and k 2 Ͻk 1 .Note that each ͱk j 2 Ϫ 2 is even in and the denominator, D͑ ͒ϭk has four simple zeros in the Riemann surface.These are located at and are not present in the first Riemann sheet.

B. Vertical electric dipole
The z ˆ-directed unit dipole is immersed in air ͑region 2, zϾ0), as depicted in Fig. 3.In the limit d→0 ϩ and z→0 ϩ the field is 17 The first Riemann sheet along with the branch-cut configuration, and the integration path are chosen as described in Sec.II A and shown in Fig. 2. Throughout the following analysis, it is assumed that ͑2.13͒

III. EXACT B 2z , E 2 , AND E 2 OF HORIZONTAL ELECTRIC DIPOLE
For mathematical convenience, consider the replacements 3. The geometry and Cartesian coordinate system for a unit vertical dipole in air.The height d is allowed to approach zero (d→0 ϩ ).
The Sommerfeld pole corresponds to q S ϭϪik S .For the purpose of carrying out the requisite integrations, q 1 and q 2 are thought of as positive with q 2 Ͻq 1 , unless it is stated or implied otherwise.The final formulas are continued analytically to complex q j ϭϪik j ( jϭ1, 2) in view of restrictions ͑2.13͒.

A. The z-component of the magnetic field
It is verified that B 2z is expressed in terms of elementary functions. 13The requisite integral equals where via the analytic continuation to ϭϪ3/2 of the right-hand side of the equation, 24 K (␣) is the modified Bessel function of the third kind. 25Hence,

͑3.4͒
It follows that This result is also derived by Wait through differentiation of the Hertz vector. 13Discussions on a similar integral appearing in the problem of the radiating vertical magnetic dipole can be found in the books by Ban ˜os 3 and Kong. 26

B. The -component of the electric field
With the definition q S ϭϪik S , consider the decomposition

͑3.9͒
The task is to express W() in terms of known transcendental functions.Following Van der Pol, 6 a first step is to convert the representation ͑3.9͒ into an integral of elementary functions.The radical in the integrand reads as follows: The interchange of the order of integration yields where, from Ref. 24 or Eq.͑A6͒ of Appendix A, Therefore, through integration by parts, where The procedure described hitherto is not different from the one in Ref. 27 for the Hertz vector of a vertical dipole.An alternative derivation of the last equation, that is amenable to generalizations, is provided in Appendix A. It is noted that W() can be expressed in terms of incomplete cylindrical functions as further discussed in Sec.VI.Despite this fact, it is more advantageous to rewrite W() as W͑ ͒ϭW͑ ;ϱ,q S ,q S ͒ϪW͑ ;ϱ,q 1 ,q S ͒ϪW͑ ;q 2 ,q S ,q S ͒.

͑3.15͒
The first term is calculated explicitly: 25 W͑;ϱ,q S ,q S ͒ϭq S ͵ 0 ϱ dy cosh ye Ϫq S cosh y ϭq S K 1 ͑ q S ͒. ͑3.16͒ 1. Integral W";ؕ,q 1 ,q S … By invoking the identity with uϭq S 2 Ϫ2 and a positive integer M , the second term in Eq. ͑3.15͒ reads as where In the above, 2 F 1 is the hypergeometric function 28 and (a) m is Pochhammer's symbol. 28y bearing in mind that 1Ϫtр͉1Ϫwt͉ for 0рtр1 and ͉w͉р1, it is inferred that for admissible complex q 1 and q 2 (Re q 1 у0), which can be used to prove the convergence of the corresponding series as M →ϱ.This relation must be supplemented with the formula in order to show that ͉R 1M ()͉ remains bounded as ͉q 1 ͉→ϱ.It is noted in passing that for m ϭ0, 1, 2, . . ., Use of the asymptotic formula (у1), in Eq. ͑3.21a͒ leads to By inspection of Eq. ͑3.14͒, the rate of convergence of the series from Eq. ͑3.18͒ is essentially independent of .In particular, U m () is approximated by This formula also holds when mϭO(1) and ͉q 1 ͉ӷ1 with Re q 1 у0, and becomes exact when mϭ0 for any q 1 .Hence, In the sense of Cauchy for convergence, The coefficients g n (q 1 ) are partial derivatives in x of the generating function, where Ei(Ϫz) is the exponential integral. 28Finally,
2. Integral W";q 2 ,q S ,q S … With the change of variable ϭvϪq S in the original integral from Eqs. ͑3.14͒ and use of the identity where now uϭ(2q S ) Ϫ1 , it is straightforward to get W͑;q 2 ,q S ,q S ͒ϭ e Ϫq S ͱ2q S ͵ 0

͑3.31b͒
In the above, one may employ the inequality ͉1ϩwt͉у1 for tу0 and Re wϾ0, to show that for complex q 1 and q 2 with Re (q 2 Ϫq S )р0, , M ϭ1,2, . . . .

͑3.36b͒
M is any positive integer.In the above, the hypergeometric functions reduce to elementary functions.For instance, by setting M ϭ1 in the second line, 2 F 1 ͑ 1 2 ,1;2;z ͒ϭ 2 1ϩͱ1Ϫz .
On the other hand, by virtue of formula ͑3.23͒, , M ӷ1.

͑3.37͒
When ͉q 2 ͉ӷ1, the corresponding sum needs to be combined with the asymptotic expansion for the modified Hankel function of Eq. ͑3.16͒, as discussed in Appendix B. In some analogy with expressions ͑3.25͒ and ͑3.26͒, which in turn leads to provided that m is a positive integer.
The aforementioned considerations indicate some rather attractive convergence properties of the series expansions when ͉q 2 2 ͉Ӷ͉q 1 2 ͉.Their termwise differentiation with respect to is legitimate and preserves the uniform-inconvergence.The series from Eq. ͑3.31b͒ is W͑;q 2 ,q S ,q S ͒ϭe i/4 q S e Ϫq 2 ͱ2q S ͚ The generating function for f m (z) is where and C(z) and S(z) are the Fresnel integrals, 28 C͑z ͒ϭ ͵ Hence,

͑3.48͒
The substitution of Eqs.͑3.8͒ and ͑3.46͒ into ͑3.7͒gives Note that when k 2 2 Ӷ͉k 1 2 ͉, the argument of each Fresnel integral becomes where ͉(k 2 Ϫk S )͉ is the Sommerfeld ''numerical distance.'' 29or k 2 ӷ1, the Hankel function in Eq. ͑3.49͒ is approximated by an expansion with the phase factor e ik S .This expansion exactly cancels terms produced by the Fresnel integrals, so that the final expression describes only waves traveling with the phase velocity of medium 1 or 2 ͑terms ϰe ik j , jϭ1, 2), as shown in Appendix B.

C. The -component of the electric field
The integral for E 2 is evaluated via the interchange of 1/ and the operator (d/d) in Eq. ͑3.7͒.The series that result through the term-by-term differentiation of expansions ͑3.27͒ and ͑3.40͒ also exhibit rapid convergence for k 2 2 Ӷ͉k 1 2 ͉, with a rate which is essentially independent of the distance .Without further ado, where W() is defined by Eq. ͑3.14͒ and the prime here denotes differentiation with respect to the argument.
It is desirable to further manipulate this formula.Decomposition ͑3.15͒ entails where With the steps of Sec.III B and for M ϭ1,2, . . ., where with g n (z) defined by Eq. ͑3.20͒, and

Exact formula for E 2
It follows that in the limit M →ϱ all series converge uniformly in .WЈ() from Eq. ͑3.52͒ reads as

͑3.63͒
Finally, substituting WЈ() in Eq. ͑3.51a͒ yields where U ˜m() and V ˜m() are given by Eqs.͑3.54͒ and ͑3.59͒.An asymptotic formula for k 2 ӷ1 can be derived along the lines of Appendix B.

A. Magnetic field
In consideration of Eq. ͑3.1͒ with q S ϭϪik S and the decomposition, Eq. ͑2.10͒ becomes From Eq. ͑3.10͒, one gets where via the analytic continuation to ϭ1 of the right-hand side of the formula 24 Alternatively, where I e (␣) is given by Eq. ͑3.8͒ and W() is defined by Eq. ͑3.9͒.B 2 can be expressed in terms of incomplete cylindrical functions. 27ith the W() introduced in Eq. ͑3.13͒, the exact B 2 from Eq. ͑4.3͒ reads as

B. The z-component of the electric field
By use of Eqs.͑4.1͒ and ͑3.10͒, Eq. ͑2.11͒ becomes After some straightforward algebra, 24 where

͑4.16b͒
As M →ϱ, the remainder R ˇ1M () approaches zero while being bounded uniformly in distance.
The rate of convergence of the exact series is independent of .

V. SIMPLIFIED FORMULAS FOR k 2 ™ͦk 1 ͦ
The exact results of Secs.III and IV are simplified considerably under the condition which holds in many cases of practical interest.In this section, connection formulas for the approximations of Appendix C are recovered to the leading order in k 2 /k 1 .

The z-component of the magnetic field
Equation ͑3.5͒ for B 2z becomes which is identical to the result given in Ref. 17 and agrees with formulas ͑C4͒, ͑C20͒, and ͑C21͒ of Appendix C. Note that condition ͑5.1͒ is redundant for establishing a smooth connection to formula ͑C4͒.

Tangential electric field
With

͑5.5͒
A close inspection of the terms inside the parentheses containing the exponential integral Ei shows that these contribute to higher orders in k 2 2 /k 1 2 .On the other hand, the Hankel function and its accompanying term are negligible for k 2 рO(1) under condition ͑5.1͒, while they are cancelled by the V m 's when k 2 ӷ1, as outlined in Appendix B. A moment's reflection leads to the uniform formula valid for all distances that are consistent with the planar-earth model.This formula yields approximation ͑C6͒ as well as ͑C24͒ and ͑C25͒ of Appendix C when k 2 Ӷ1 and k 2 ӷ1, respectively.Similar steps can be taken for E 2 of Eq. ͑3.64͒, to obtain In consideration of the asymptotic expansion ͑B18͒ of Appendix B, it is inferred that This formula is useful for all reasonable purposes yet it assumes that which poses no practical restriction.The formula agrees with approximations ͑C5͒, ͑C22͒, and ͑C23͒ of Appendix C.

B. Vertical electric dipole 1. Magnetic field
Equation ͑4.6͒ for B 2 furnishes Under the sensible condition the exponential integral and its accompanying term can be neglected.Consequently, which connects smoothly to expressions ͑C10͒, and ͑C32͒ and ͑C33͒ of Appendix C.

The z-component of the electric field
The retainment of the first term in each series of Eq. ͑4.22͒ for E 2z yields With condition ͑5.11͒, the preceding expression becomes in agreement with approximations ͑C11͒, and ͑C34͒ and ͑C35͒ of Appendix C. Approximations ͑5.6͒, ͑5.8͒, ͑5.12͒, and ͑5.14͒ are in full agreement with the formulas of King et al., 17 provided that the replacement of (k 2 Ϫk S ) by ဧ is made according to ͑3.50͒.

VI. CONCLUSIONS AND DISCUSSION
We start this paper with the Fourier-Bessel integral representations for the fields in the planar boundary between air and a homogeneous half space of infinitesimal electric dipoles lying in the interface.The focus is on the components E and E of the horizontal dipole and B and E z of the vertical dipole in the cylindrical coordinates of Figs. 1 and 3.][6][7][8][9][10] The present analysis is believed to go a step further by relaxing the condition k 2 2 Ӷ͉k 1 2 ͉ and replacing the integrals by simple, exact integrated series which are usable for any distance from the source.It is verified that the B z component of a horizontal dipole is described by simple elementary functions.
The exposition bears two appealing features.The first feature is that the ratio of any successive terms in each series is shown to be proportional to k 2 2 /k 1 2 , i.e., the inverse of the diffraction index squared, while it remains bounded uniformly in .The relative errors due to the retainment of a finite number of terms in the series, say M , are essentially of the order of (k 2 2 /k 1 2 ) M regardless of k 2 and k 1 .In most cases of practical interest where ͉k 1 ͉у3k 2 , at most three or four terms of each expansion suffice for reasonable accuracy.
The second feature is that the summands are expressed in simple closed form as the wellknown exponential and Fresnel integrals.These functions explicitly reveal the dependence on the physical parameters such as the k 1 and the Sommerfeld numerical distance.They also provide a natural connection to the recently obtained, approximate formulas of King et al. that distinguish between the direct and ideal-image fields and the lateral-wave or surface-wave contributions when k 2 2 Ӷ͉k 1 2 ͉. 17 The present treatment not only verifies the results of these authors by different, exact means, but also extends their validity to distances close to the source.
The two features mentioned above illustrate the advantages of the proposed formulas over representations of the incomplete Hankel function used for the same purpose. 27The price that one seems to pay for this simplicity, however, is the limitation in the choice of possible configurations or field components that can be treated exactly in a similar fashion. 21Obtaining integrated series of analogous properties for the remaining components of Eqs.͑2.1͒-͑2.6͒and ͑2.10͒-͑2.12͒ is an open problem for future work.
Higher-order terms of the derived series may become of importance for radiowave propagation over a very dry earth; another example of applications could perhaps be related to the so-called ''low-k'' dielectric insulators. 30As in Ref. 19, the present model is restricted in its applicability due to the assumption of a planar boundary.][33][34] W͑ ͒ϭ q 2 q 1 I͑q 1 ,q S ;Ϫ1/2͒Ϫ q 1 q 2 I͑q 2 ,q S ;Ϫ1/2͒, ͑A1͒ attention is focused on the integral Of course, I is understood as Abel summable in x→ϱ.Without loss of generality, ␣ and ␤ are assumed to be positive.The inequality ␣у␤ is imposed for definiteness.

Case ␣Ä␤
With ␣ϭ␤, The starting point is the known formula 24 where K Ϫ is the modified Bessel function of the third kind.Note that one may not set ϭϪ1 on both sides of this equation simultaneously.Caution needs to be exercised because allowing →Ϫ1 ϩ in results in a nonintegrable singularity at xϭ0 with a vanishing numerical coefficient.

Let
Aϭx 2 ϩ␣ 2 , Bϭx 2 ϩ␤ 2 , ͑A8͒ and consider the integral representations The radical in Eq. ͑A2͒ is recast in the form Analytic continuation to complex with Re Ͻ0 is brought about via the integral where C is a closed contour in the u-plane.C originates from uϭ1 in the first Riemann sheet and encircles the origin in the clockwise sense, as shown in Fig. 4. The first Riemann sheet is defined so that u Ϫ1ϩ Ͼ0 and ͓AuϩB͑ 1Ϫu ͔͒ Ϫ1Ϫ Ͼ0, 0ϽuϽ1, ͑A12͒ with the associated branch cuts lying along the positive and negative real axis.Note that in addition to the branch point at uϭ0 another branch point exists in the negative axis at uϭ ϪB/(AϪB).With Eq. ͑A6͒, it follows that I͑␣,␤; ͒ϭ Hence, I(␣,␤;) is an integral of an elementary function if ϭnϪ1/2, n: integer.Let tϭt͑u ͒ϭͱ␣ 2 uϩ␤ 2 ͑ 1Ϫu ͒, t͑1 ͒ϭ␣, t͑0 ͒ϭ␤.͑A14͒ This transformation maps C onto CЈ ͑Fig.5͒:

͑A15͒
By virtue of the identity application of integration by parts to Eq. ͑A15͒ furnishes

͑A20͒
Since the integrand has now an integrable singularity at tϭ␤, the path can be indented back to the positive real axis:

͑A21͒
This result agrees with Eqs.͑3.13͒ and ͑3.14͒.Evidently, applying Eq. ͑A18͒ does not produce any new integrals of elementary functions.

Wave through region 2
When ͉q S ͉ӷ1, the major contribution to integral ͑B2͒ arises from the vicinity of the lower endpoint of width O͓(q S ) Ϫ1 ͔.If in addition ͉(q 2 Ϫq S )͉рO(1), ϭ0 falls inside the critical region and the radical can be replaced by a Maclaurin expansion.Accordingly, where

͑B10͒
Of course, in the limit M →ϱ the remainder ⌼ M () is unbounded and the series from Eq. ͑B3͒ diverges.In the sense implied by Eq. ͑B3͒, W͑;ϱ,q 2 ,q S ͒ϳϪe Ϫq 2 q S ͚ mϭ0 ϱ V m ͑ ͒. ͑B11͒ This asymptotic expansion can be attained somewhat heuristically from the exact series ͑3.40͒ combined with the asymptotic expansion for H 1 (1) (k S ).Notice that dz m ͓z Ϫ1/2 e iz F͑z ͔͒.͑B12͒
Accordingly, expand the radical as and, for nϭ1, 2, . . ., 28 A ͓x͔ denotes the integral part of x.Note that the second line of Eq. ͑B16͒ holds even for nϭ0.It follows that The first term inside the braces is the U 0 () from Eq. ͑3.19͒.This expansion can be verified directly from Eq. ͑3.18͒ by invoking the formula 28 which holds uniformly in Arg z, interchanging the order of summation and subsequently allowing M →ϱ.

Asymptotic formula for E 2
The resulting asymptotic expansion for E 2 reads as ͑B19͒ Similar expressions can be written down by inspection for the other components of Secs.III and IV.

A. Horizontal electric dipole
With these approximations, B 2z becomes while the integral I m is given by Eq. ͑3.4͒.Hence, In the same vein, by virtue of Eq. ͑3.8͒ for I e ,

͑C6͒
The computation of the limiting forms of E 2z , B 2 and B 2 is more involved: where use is made of Ref. 36, and Compare with the exposition in Ref. 13.In principle, the behavior of the z-component of the electric field depends on the limit path.For instance, if the observation point is forced to approach the source along a straight line from region 1 (zϽ0), this component behaves as ϳ1/ 3 .͓The divergence can ensue from taking ⑀ϭ in the second line of Eq. ͑C7͒.͔
Following Sommerfeld, 29 one may replace each J n (nϭ0, 1, 2) by (1/2)͓H n (1) ϩH n (2) ͔.The contour ⌫ can be chosen symmetric under inversion through the origin, as shown in Fig. 2. Use is made of the analytic continuation formula

͑C17͒
where the upper sign holds along the left-hand side and the lower sign along the right-hand side of each branch cut.For instance, D() of Eq. ͑2.8͒ becomes D͑ ͒ϳϮe Ϫi/4 ͱ2tϩk 2 /k 1 , ϳk 2 ,

͑C19͒
In each F 2, j , the contribution from the circle C ␦, j of radius ␦ vanishes in the limit ␦→0 ϩ .

A. Horizontal electric dipole
The integrals for the z-component of the magnetic field are

FIG. 2 .
FIG.2.Branch-cut configuration and integration paths pertaining to the Sommerfeld integrals ͑2.1͒-͑2.6͒for the horizontal electric dipole and ͑2.10͒-͑2.12͒for the vertical electric dipole.The original integration path is shown with arrows in the positive real axis.The contours ⌫ and ⌫ j ( jϭ1,2) serve the asymptotic evaluations for k 2 ӷ1 carried out in Appendix C.

FIG. 4 .
FIG. 4. Contour of integration C for the integral of Eq. ͑A11͒.
͑C23͒ ဧ and F(z) ͑C25͒The rest of the components are calculated under similar approximations as follows:͑C31͒Compare with Ref. 17.