1. Introduction

This review article is concerned with exhibiting techniques leading to either closed form expressions for Kapteyn series or integral representations that cannot be further reduced.

In general there are two sorts of Kapteyn series [1]. Kapteyn series of the first kind are infinite sums of Bessel functions of the form

that is, Kapteyn series of the first kind involve summations over terms containing one Bessel function of the form , while Kapteyn series of the second kind involve terms each of which is proportional to a product of two such Bessel functions. Note that the index of summation appears both in the order and in the argument of the Bessel functions.

Kapteyn series arise in a host of mathematical physics problems. The range extends from pulsar physics [2, 3] through radiation from rings of discrete charges [4, 5] through quantum modulated systems [6, 7] through traffic queuing problems [8, 9] and on to plasma physics problems in ambient magnetic fields [10, 11] to name but a few such disciplines. Therefore, it seems appropriate to spell out a variety of techniques that can be used separately or in combination to sum such series efficiently.

While some procedures for summation of selected Kapteyn series in mathematical physics have been known for over a century, the purpose here is to provide more general methods of broad use for many categories of such series. This purpose is based on many physical applications that have arisen over the last half century where, to date, either only asymptotic representations of the relevant Kapteyn series have been given or where recourse to direct numerical investigations have been given without considering whether closed form expressions exist at all for the series.

In the latter situation it is difficult to determine whether the numerical methods provide accurate results because one has no basic template in closed form (or at worst in integral form) against which a comparison can be made. In the former case, while it is often that one can compare a known asymptotic representation of a Kapteyn series against numerical results, often one does not know the domain of validity of the asymptotic expansion nor does one know the functional behavior of the Kapteyn series in regions removed from the asymptotic result nor, indeed, does one have available the general domain of convergence of the desired Kapteyn series.

For all of these reasons it is appropriate to review some general methods that can be used to sum a large array of Kapteyn series in mathematical physics.

2. Kapteyn Series in Pulsar Radiation Problems

In discussing radiation in vacuum from a rotating magnetic dipole, which is off-center with respect to a rotating pulsar, but which is “frozen” in the pulsar body, Harrison and Tademaru [2] showed that the total power radiated, , is given by

and the force, , acting on the dipole in the direction is

where are the magnetic dipole components in a cylindrical coordinate system (see Figure 1), is the angular velocity of the pulsar, with being the offset distance of the dipole from the spin axis, and where

Figure 1: Sketch of the spin and dipole coordinates from Harrison and Tademaru [2].

Note that is required so that the series and are convergent.

Harrison and Tademaru [2] argued that for values of one could approximate the power, , and the force as given in their Equations () and (). However, the fact that is in the range means that it is not easy to justify their expansion procedure. Further, in situations where a pulsar has a high spin rate and where the offset distance can approach the radius, , of the pulsar, the factor is so that is almost nowhere valid. To investigate such situations one needs closed form expressions for the two series and . Watson [12] refers to these series as Kapteyn [1] series of the second kind, which series have been investigated to some extent by Nielsen [13].

This section provides the general procedure for evaluating the series (2.3) and (2.4), although the method is of much greater generality as it will become clear in the course of its development.

2.1. Manipulations with the Series and

First, differentiate with respect to to obtain

Use Bessel's equation (e.g., [14, Section ]) for in the form

in (2.5) to obtain

so that

because the integration constant in (2.8) is zero by evaluation as .

Thus, it is sufficient to evaluate in closed form and to perform the integration in (2.8) to obtain .

Consider then . Use the formula [15, Section ]

in the form (e.g., [16, Section ])

so that

Expression (2.11) shows that is expressed as an integral over a Kapteyn series of the first kind, for which several theorems are available as expressed in [15]. The most important result needed is the following.

If the Kapteyn series

where is arbitrary but given, is known in closed form, then the series

is given by two simple integrations because

by direct differentiation of (2.13). Again, is required so that the series is convergent. Furthermore, the differential operator in (2.14) with respect to an arbitrary variable is introduced as

Reversing the argument: if is known, then is given directly by differentiation of in (2.14).

2.2. Reduction of to Closed Form

From (2.11)–(2.14) we have that (with if is odd, and if is even)

where . But it is well known that [15, Section ]

where , so that, also for ,

Hence

Carrying out the differentiations in (2.19) yields

with .

Using a partial fraction expansion and

where , and using

the integral in (2.20) can be completed in closed form yielding

Inserting this expression for into (2.8) and performing the integral leads to

Thus, this procedure shows that both and are available analytically. Numerical comparison of direct series evaluation term by term with the closed form analytical expressions confirms agreement to at least one part in . The prototype of such Kapteyn series of the second kind was first given in closed form by Schott [17], who evaluated

(Note that, in (2.25), there appears a factor which was missing in Lerche and Tautz [3].)

The basic procedure for evaluating Kapteyn series of the generic form

where is either an integer or half integer (positive or negative) and is either unity or , then follows the same recipe as given here, although the expressions rapidly become unwieldy as becomes large.

2.3. Calculation of and

The closed form expressions for and can then be used in (2.1) and (2.2) to evaluate the radiated power and force on the dipole in terms of . The result leads to elliptic integrals which cannot be solved analytically. But an expansion of the result in powers of yields and , where

Note that the zero-order terms and agree with the expansion given by Harrison and Tademaru [2] in their Equations () and ().

Next, the expression for is separated as

and the functions , , and as well as are calculated numerically. Comparing the exact function values to the approximations from (2.27), drastic deviations are revealed even from the first-order approximations, as illustrated in Figures 2 and 3. The relative deviations are shown to reach 10% even for as low as (zero-order approximation) and (first-order approximation).

Figure 2: Comparison of the exact and approximate values for the three components of the function . In panels 1, 3, and 5, the solid lines show the (numerically calculated) exact functions , , and , respectively, and the dashed and dash-dot lines show the approximations and , respectively. All function values are normalized to . In panels 2, 4, and 6, the relative deviation (in percent) from the exact function values is shown for the approximations (solid lines) and (dashed lines), respectively. The two dotted lines mark deviations of and .

Figure 3: Comparison of the exact and approximate values for the function , normalized to . In the upper panel, the solid line shows the (numerically calculated) exact function , and the dashed and dash-dot lines show the approximations and , respectively. In the lower panel, the relative deviation (in percent) from the exact function values is shown for the approximations (solid line) and (dashed line), respectively. The two dotted lines mark deviations of and .

The expansion parameter , however, is normally very small as will be illustrated by two examples: (i) the fastest rotating pulsar [18] (PSR J1748–244ad) has a rotation period of s with a radius of 8 km and, therefore, the offset of the dipole from the spin axis, , would have to be as large as  km in order to have , with these parameters one would have obtained a deviation of resulting from the approximations of Harrison and Tademaru [2]; (ii) for the Crab pulsar [19] (PSR B053121) the parameter yields with the pulsar radius, which is small even for large offsets .

If the surface velocity approached the speed of light, the expansion parameter would be given by (without taking into account any relativistic effects); thus, can, at least in principle, attain values where both the zero-order and the first-order approximations from (2.27) become invalid.

3. Kapteyn Series in Other Radiation Problems

One problem in radiation that was considered of great interest at the beginning of the 20th Century is the following. It is well known that a single point charge, moving uniformly in a circle, radiates. Suppose then that one has charges equally spaced around a circle and all moving at the same circular speed. Then they, too, radiate. Now as the number of charges is increased, all other conditions being held fixed, then the spacing between charges decreases proportional to . The limit of this process is a continuous uniform charge distribution moving with constant circular motion, that is, a steady-state ring current. But it is also well known that such a current formation does not radiate. Then the question is as how does the radiation diminish so that, finally, there is no radiation from a continuous ring current?

Investigations of this basic problem immediately encountered Kapteyn series of the second kind (see, e.g., [1, 15]) in a variety of forms and guises. While the formula describing the radiation output was expressible as a set of terms involving sums of Kapteyn series, at first only approximations to the series could be obtained for arbitrary [4]. The work of Budden [20] provided a systematic determination of the Kapteyn series involved and evaluated the radiation field of the like particles in terms of factors summed to . The advantage was that, along the way, Budden managed to effect solutions in closed analytical form to some of the Kapteyn series involved. The upshot was that, as , one could show how the radiation field diminished to zero.

Since that time there has been, and continues to be, interest in a variety of such radiation types of problems. Alternating positive and negative point charges spread uniformly around a ring, each of which moves at constant circular speed, is one such problem [17]. As the number of charges increases without limit the spacing between successive charges tends to zero so that, in the limit, there is a charge neutral ring that does not radiate. The approach of the radiation field to zero as the number of charges tends to infinity is the problem of interest. Fortunately this problem is just a variant of the problem solved by Budden [20] because it represents two rings of opposite charges with twice the spacing. Budden's solution is then immediately appropriate by superposition and charge reversal.

Radiation from a magnetic dipole, off-center from a pulsar that spins, is another such problem, as we have seen earlier in this review [2, 3], as is the radiation field from a charged particle undergoing elliptical motion [21].

In all such problems there have arisen, to date, twelve basic Kapteyn series of the second kind, some of which have been known in closed form for a while while others are often referred to as “solved” but seem to be not readily available, if at all.

The next section of the review provides the basic methodology to handle all twelve of the series and shows which are expressible in closed analytic form, and which are only expressible only as integrals that cannot be reduced to analytic form.

3.1. Manipulations with Basic Sets of Kapteyn Series

3.1.1. The Sets of Series

The twelve series in question are given by

where and .

Determination of the sets of series can be reduced to the simpler problem of determining only the set of series with (in the cases of , , and ) and the set of series with (in the cases of , , and ).

The reason for these reductions is as follows. One can write

so that it is sufficient to obtain , , , and .

Note also that

But, because of Bessel's equation (see (2.6) with changed to ), one has

so that

Equally

Thus it is sufficient to obtain and .

One can also use the theorem due to Watson [12] of (2.14), which was derived in Section 2.1, and which yields if is known. Alternatively, if is known then is given by direct differentiation.

Consider then . Use (2.10) so that

But the series

is precisely of the form required in Watson's theorem, with if is odd and if is even, so that

where the differential operator from (2.15) has been used. Hence, for all series of the type can be reduced to the determination of by differentiation. Equally, for one can use Watson's theorem in the converse sense to note that

so that, by two integrations, one has a recursive relation leading directly to .

Thus, all twelve of the basic series needed can be written in terms of four fundamental series

for . All other series (with , or , resp.) are directly given as simple differentials or simple integrals with respect to of one or the other of the four fundamental series. It is, therefore, both necessary and sufficient to consider and .

3.1.2. The Two Series Represented by

Set

Now, in , replace the Bessel functions using again (2.10) while in replace and

Then write

(see [15]).

In principle, one could also use a representation of the Bessel function in exponential form [16] see and then carry out the summation. However, because (3.12a) and (3.12b) are a product of two Bessel functions, this ansatz would be even more difficult than the approach followed here.

Now, inserting (2.10) and (3.14) into expression (3.12a) for and inserting (3.13) and (3.15) into expression (3.12b) for and then performing directly the infinite sums lead, after some tedious but elementary algebra, to

Numerical investigation by direct summation of and as given in (3.12a), (3.12b) and comparison with the simple integral formulations given in (3.16a), (3.16b) shows that the series are indeed given by (3.16a), (3.16b) to better than a part in ; this limit on resolution being caused by numerical round-off error. Figures 4 and 5 show the comparison between the integrals and direct summation as a function of increasing for both and , respectively, with the relative error (in percent) also being plotted. (Note that, for numerical reasons, the relative error increases above percent as (Figure 4) and as (Figure 5), respectively. Such depends heavily on the numerical summation and integration methods as well as on the computer time. By expansion of the integrals around and , however, one can get almost exact agreement of the series and the integral.)

Figure 4: The series from (3.12a) with the relative error when compared to the integral from (3.16a).

Figure 5: The series from (3.12b) with the relative error when compared to the integral from (3.16b).

Throughout this review, the numerical evaluation of infinite sums is carried out as follows: First, a number of terms (usually ) is summed directly; to accelerate the convergence of the sum, then Wynn's epsilon method (see, e.g., [22, 23]) is used, which samples a number of additional terms (usually ) in the sum, and then tries to fit them to a polynomial multiplied by a decaying exponential. Thus, the series are well approximated and the required computer time is kept moderate. The convergence of the sums, in addition, is guaranteed by analytical considerations. Furthermore, numerical integrations are carried out using standard techniques such as adaptive grids. However, some care has to be taken of the square-root singularity (e.g., at in (3.16a) and (3.16b)). Since we used Mathematica version 6.0, this problem is dealt with automatically. Using other packages, however, appropriate measures would have to be taken manually.

Marshall [21] suggested that the sum , written in the form

could be represented by a single elliptic integral (his (2.22)) as

Figure 6 shows plots (as a function of ) of both the sum and the elliptic integral representation, from (3.18), suggested in [21]. There is no agreement even at the crudest level of approximation that indicating the elliptic integral is not appropriate.

Figure 6: The series from (3.17) (solid line) compared to the integral representation from (3.18), as given in Marshall [21] (dashed line). In the lower panel, the relative error with respect to the direct summation of the series is shown.

3.1.3. The Two Series Represented by

Set

The series has been known in closed form since the time of Schott [17]. Use the well-known fact [1] that

Integrate (3.20) over , thereby obtaining

which is just Schott's [17] formula.

The series is considerably more complicated to evaluate. Write

Now use the Schlömilch [24] formula, which states that any function

which is given through an arbitrary (but known) function , can be rewritten as

Set with so that

With the identifications and , (3.22) then yields

where the upper integration limit is implicitly given by , or is given explicitly by . One can then write

which might be more amenable when numerical integration is required. Figure 7 compares given by (3.25a) with direct term by term summation of the series in (3.19b), showing that, to within about 1 part in , the two are identical in the interval (cf. footnote). Note also that the integral representation of is convergent for all values of , including .

Figure 7: The values for as a function of (a) and the series from (3.19b) together with the relative error when compared to the integral from (3.25a) ((b), (c)).

3.2. Discussion

A general method has been presented for the evaluation of twelve Kapteyn series of the second kind. Such series are important for the analytic description of radiation processes in various astrophysical applications such as the radiation from off-centered dipoles in neutron stars. Originally, the Kapteyn series described here arose when the attempt was made to describe the radiation from a distribution of a finite number of discrete point charges, all moving at uniform spacing at constant speed in a circle.

Previously, most of the Kapteyn series have not been evaluated or, in the case of one of the series, were written in terms of a single elliptic integral, which turned out to be invalid when evaluated numerically (see (3.18)). Equation (3.25a) is more appropriate because it represents the series in terms of a different, but also elliptic, integral.

As has been shown here by recurrence relations, there are only four basic series that need to be calculated, one of which was already known in closed algebraic form. All other of the twelve series can be obtained from direct differentiation or integration of one or other of the four basic series. The series can be evaluated in terms of closed analytic expressions or in terms of integrals that cannot be further reduced. Numerical calculations were carried out to compare the values obtained by direct summation to those obtained from the integral representations, and the relative errors (less than a part in ) were shown to be limited by numerical round-off errors that are responsible for the differences occurring between direct series representations and integral representations of the series.

Furthermore, the method presented here may be useful when one has other Kapteyn series of the second kind to consider, thereby providing an additional reason to consider such series anew.

4. Kapteyn Series in Quantum-Modulated Systems

Kapteyn series of the second kind also appear in models of even- and odd-order sideband spectra in the optical regime of a quantum system modulated by a high-frequency (e.g., terahertz) electromagnetic field [6] and in certain time-periodic transport problems in superlattices [25, 26]. This section shows that both the even- and the odd-order Kapteyn series that appear can be summed in closed form, thereby allowing more transparent insight into the structural dependence of the sideband spectra and also providing an analytic control for the accuracy of numerical procedures designed to evaluate the series (see also [7]).

In discussing an optical analogue for phase-sensitive measurements in quantum transport through a quantum dot whose energy levels are modulated periodically in time, Citrin [6] has considered optical propagation of a monochromatic optical beam at frequency (known as the fundamental frequency) transmitted through or reflected from a quantum well modulated by a high-frequency field (henceforth called the terahertz field) at frequency . The transmitted and reflected optical beams are shown to contain new frequencies where is an integer, known as terahertz sidebands [6]. The amplitude of such signals as a function of is known as terahertz sideband spectra. In the limit that only one modulated energy level (at time-averaged energy ) is relevant and the periodic modulation of that energy level is sinusoidal, a simple and useful model can be obtained that permits considerable analytic progress to be made before numerical methods need to be brought to bear on the problem. Such a model then permits one to study in a straightforward fashion how the terahertz sidebands scale with various parameters such as and the modulation strength (the degree to which the energy level varies with respect to its time average ).

A formally similar analytic model also arises in connection with miniband transport in a superlattice subjected to a strong terahertz field [25, 26]. The phases of the reflected and transmitted complex electromagnetic amplitudes for each sideband (with respect to the initial optical beam at angular frequency ) provide information on the quantum system. The detailed development given by Citrin [6] has its basic underpinning from the calculation of the amplitude of the transmitted optical electric field, , at frequency . Equation (2.1) of Citrin [6] provides

with

where

The series is the Kapteyn series of the second kind of interest here. The notation in (4.1) through (4.3) is that given by Citrin [6]. In particular, the prime on the summation indicates that only terms where the parity of is that of are retained and is the (sideband order -dependent) detuning between the average energy of the fundamental and relevant sideband and the time-average energy of the modulated level . The first term in (4.1) gives the transmitted beam at the input frequency in the absence of the modulation field, while the second contains the terahertz sidebands at . The cardinal point for this section is the requirement that the sum in (4.3) is the sum over integers with the same parity as . Thus if (with ) then (with ), while if then (with ). Note that due to the form of (4.3), there is no need to consider negative values of . Citrin [6] notes that by expanding (4.3) in powers of one can identify the various multiphoton processes contributing to each sideband, and he provides the appropriate expansion. Numerical evaluation at this stage is required and has the consequence that convergence of an infinite product inside an infinite sum must be proven, a less than trivial task.

The purpose here is to show that the Kapteyn series represented in (4.3) can indeed be summed in closed form, thereby facilitating not only the general understanding of the sideband spectra but also obviating the need to prove convergence of an infinite product inside an infinite sum—a serendipitous result that is definitely a welcome blessing. Moreover, the closed-form expressions found as well as the approach by which they are obtained are likely to be of interest for other areas of physics and applied mathematics.

4.1. Evaluation of the Kapteyn Series

For (and so ), that is, for the even-order sideband spectra, one has to evaluate

with , for all nonnegative integers .

For (with ) and so (with ), that is, for the odd-order side spectra, one has to evaluate

with , for all integers including <