Kapteyn Series Research Articles

On a special Kapteyn series

We investigate the mathematical properties of the function , ϵ ∈ [−1, 1], which is a special Kapteyn series of the first kind. Unlike various other special Kapteyn series, the function T (ϵ) does not seem to possess a closed-form expression. We derive an integral representation for T (ϵ) from which various properties of T (ϵ) can be established. In particular, monotonicity and convexity properties of T (ϵ) and ϵ−1 T (ϵ) can be shown. Also, the behaviour of T (ϵ) as ϵ ↑ 1 can be determined from the integral representation. Furthermore, while the Kapteyn series representation of T (ϵ) is very slowly convergent when ϵ is close to ±1, a regularized form of the integral representation of T (ϵ) allows to compute T (ϵ) accurately using Simpson’s rule with relatively few sample points of the involved integrand.

On the Bessel Solution of Kepler’s Equation

Since its introduction in 1650, Kepler’s equation has never ceased to fascinate mathematicians, scientists, and engineers. Over the course of five centuries, a large number of different solution strategies have been devised and implemented. Among them, the one originally proposed by J. L. Lagrange and later by F. W. Bessel still continue to be a source of mathematical treasures. Here, the Bessel solution of the elliptic Kepler equation is explored from a new perspective offered by the theory of the Stieltjes series. In particular, it has been proven that a complex Kapteyn series obtained directly by the Bessel expansion is a Stieltjes series. This mathematical result, to the best of our knowledge, is a new integral representation of the KE solution. Some considerations on possible extensions of our results to more general classes of the Kapteyn series are also presented.

Addendum to “Poroacoustic solitary waves under the unidirectional Darcy–Jordan model” [Wave Motion 94 (2020) 102498

We revisit the poroacoustic studies carried out in Jordan (2020) with the aims of extending, classifying, and clarifying the results presented therein. In the process, connections to celestial mechanics, Kapteyn series, and quantum mechanics are noted.

A Jentzsch-Theorem for Kapteyn, Neumann and General Dirichlet Series

Comparing phase plots of truncated series solutions of Kepler’s equation by Lagrange’s power series with those by Bessel’s Kapteyn series strongly suggests that a Jentzsch-type theorem holds true not only for the former but also for the latter series: each point of the boundary of the domain of convergence in the complex plane is a cluster point of zeros of sections of the series. We prove this result by studying properties of the growth function of a sequence of entire functions. For series, this growth function is computable in terms of the convergence abscissa of an associated general Dirichlet series. The proof then extends, besides including Jentzsch’s classical result for power series, to general Dirichlet series, to Kapteyn, and to Neumann series of Bessel functions. Moreover, sections of Kapteyn and Neumann series generally exhibit zeros close to the real axis which can be explained, including their asymptotic linear density, by the theory of the distribution of zeros of entire functions.

Improved Massive MIMO RZF Precoding Algorithm Based on Truncated Kapteyn Series Expansion

In order to reduce the computational complexity of the inverse matrix in the regularized zero-forcing (RZF) precoding algorithm, this paper expands and approximates the inverse matrix based on the truncated Kapteyn series expansion and the corresponding low-complexity RZF precoding algorithm is obtained. In addition, the expansion coefficients of the truncated Kapteyn series in our proposed algorithm are optimized, leading to further improvement of the convergence speed of the precoding algorithm under the premise of the same computational complexity as the traditional RZF precoding. Moreover, the computational complexity and the downlink channel performance in terms of the average achievable rate of the proposed RZF precoding algorithm and other RZF precoding algorithms with typical truncated series expansion approaches are analyzed, and further evaluated by numerical simulations in a large-scale single-cell multiple-input-multiple-output (MIMO) system. Simulation results show that the proposed improved RZF precoding algorithm based on the truncated Kapteyn series expansion performs better than other compared algorithms while keeping low computational complexity.

Analytical Determination of Conduction and Switching Power Losses in Flying-Capacitor-Based Active Neutral-Point-Clamped Multilevel Converter

Multilevel converters are mainly used in medium-voltage high-power applications. Active neutral-point-clamped (ANPC) flying capacitor multicell (FCM) converter is a well-known type of multilevel converters which is commercially available in high-power medium-voltage motor drive market. Since power loss investigation can be very advantageous in the design phase of multilevel converters, this paper presents an analytical approach to calculate and investigate the conduction and switching power loss in ANPC-FCM converter. First, the RMS and average currents of insulated-gate bipolar transistors (IGBTs) and antiparallel diodes are analytically calculated by considering the associated duty cycle of each IGBT and diode, converter modulation index, load current, and load power factor. Numerical results of the derived closed-form equations to calculate the RMS and average currents of IGBTs/diodes are compared with simulation results and experimental measurements. Numerical results match the simulation results and experimental measurements which validates the derived closed-form equations. Afterward, the obtained equations for RMS and average current computations are utilized to calculate the conduction power losses in a 12.1-MVA 6.6-kV nine-level (line-to-line) ANPC-FCM multilevel converter. For this purpose, a 4.5-kV 1.2-kA IGBT module from ABB is considered as a power switch and its parameters are employed in analytical computations and simulation of the ANPC-FCM multilevel converter for conduction power loss determination. Moreover, closed-form equations are derived for analytical determination of switching power losses for ANPC-FCM converter using Kapteyn (Fourier-Bessel) series. Based on the derived closed-form equations for conduction loss and switching loss calculation, a method is presented to determine the junction temperature in IGBTs and diodes for ANPC-FCM converter.

Kapteyn series and photon emission

Sums of Kapteyn series of several types are obtained and relations between them are found. Special attention is paid to series whose sums are expressed in terms of transcendent functions. Their asymptotics are determined. In some cases, integral representations were obtained and reduced to the form convenient for numerical calculations. The electron lifetime in a constant electromagnetic field before photon emission is determined. It was indicated that this time increases with the ultrarelativistic electron energy in a strong constant field.

Detailed and comprehensive mathematical modeling of flying-capacitor stacked multicell multilevel converters

Purpose – This study aims to propose a mathematical model for stacked multicell converters (SMCs), to be exploited in the analytic determination of natural voltage balancing dynamics of the flying-capacitor (FC) stacked multicell multilevel converters, i.e. investigations of the start-up behavior, dynamic response, and natural voltage balancing phenomenon. Design/methodology/approach – The crux of the proposed strategy is based on the closed-form analytic solution derivation for the switching functions used in the switching of the SMCs operated under phase disposition (PD) and phase shifted carrier (PSC) pulse width modulation (PD-PSC-PWM) technique. Hence, the suggested approach develops an analytic solution for the Fourier series and associated Fourier coefficients pertinent to the switching functions of the SMCs by obtaining the switching instants of the PD-PSC-PWM modulator in terms of Kapteyn series when the frequency of the triangular carrier waveform (fc) and that of the sinusoidal reference waveform (fr) have an integer ratio, i.e. f c · f r −1=k, k∈N. Findings – This approach results into a model, first order differential equation based model, which can be readily developed for the SMCs with any number of levels expediting the investigation of their performance. Furthermore, by an experimental scrutiny conducted on a 4×2-cell-nine-level topology of an SMC, it is inferred that under PD-PSC-PWM modulation technique, FC voltages balance naturally for higher number of stacks and cells, therefore the natural balancing exist for high-level SMCs. Research limitations/implications – Despite the sophistication of the proposed methodology and mathematical model, this study presents an alternative approach with high potential of applicability for derivation of the multilevel converter mathematical model exploiting the Kapteyn (Bessel-Fourier) series. Practical implications – Numeric computation results of the proposed analytic model for the SMCs and the simulation results as well as investigational measurements taken from 2×2-cell-five-level and 4×2-cell-nine-level experimental set-ups are presented in order to substantiate the suggested approach, derived model, and verification of natural balancing. Originality/value – This article and its innovations are original.

Analytical modelling of single‐phase stacked multicell multilevel converters exploiting Kapteyn (Fourier–Bessel) series

This study proposes a mathematical model for stacked multicell (SM) converters (SMCs), to be exploited in the analytic determination of the natural voltage balancing dynamics of the SMCs, that is, investigation of the start-up behaviour, dynamic response and natural voltage balancing phenomenon. The crux of the proposed strategy is based on the closed-form analytic solution derivation for the switching functions used in the switching of the SMCs operated under phase disposition (PD) and phase-shifted carrier (PSC) pulse-width modulation (PD–PSC-PWM) technique. Hence, the suggested approach develops an analytic solution for the Fourier series and associated Fourier coefficients pertinent to the switching functions of the SMCs by obtaining the switching instants of the PD–PSC-PWM modulator in terms of ‘Kapteyn series’ when the frequency of the triangular carrier waveform (f c) and that of the sinusoidal reference waveform (f r) have an integer ratio, that is, f c f r −1 = k, k ∈ ℕ. This strategy results into a model (‘first-order differential equation based model’) which can be readily developed for the SMCs with any number of levels expediting the investigation of their performance. Numeric computation results of the proposed analytic model for the SMCs and simulation results as well as measurements taken from an experimental set-up are presented in order to validate the suggested approach and derived model.

DC (direct current) voltage source reduction in stacked multicell converter based energy systems

Multilevel power converters are key elements in the electrical grid-tied energy systems. This paper presents a novel topology for stacked multicell (SM) multilevel converters. The suggested SM converter has considerable interests of applications for interconnection of energy systems (renewable energy generation systems, energy storage systems, HVDC (high voltage direct current) systems, FACTS (flexible alternating current transmission system) devices, and custom power devices) to the electric power systems. The main advantages of the proposed converter, compared to the conventional SM converter, for attaining the aforementioned purpose are the reduction of the DC voltage energy sources from two to one, power switches (IGBT's) voltage rating, converter cost and size, losses, and installation area.The proposed SM converter has been analytically modeled by obtaining the switching instants of the PWM in terms of the Kapteyn series. Numerical solution results for the derived model of the proposed converter, simulation results, and measurements taken from an experimental set-up are presented in order to validate the operation and performance of the suggested multilevel topology.The proposed SM converter has been exploited in a dynamic voltage restorer (DVR). This proposed custom power device is based on the one energy storage device (battery, capacitor, super capacitor) with halved voltage rating owing to utilizing the suggested SM multilevel inverter (DC/AC converter) as an interface between its energy storage component and the electrical grid. Simulation results confirm the capability of the suggested energy storage based grid-tied custom power device (DVR) in solving power quality problems.

Elimination of one dc voltage source in stacked multicell converters

This study presents a novel configuration for stacked multicell (SM) converters. The main advantage of the proposed converter, in comparison with the conventional one, is that the number of required dc voltage sources is reduced from two to one in the proposed topology which results in decreasing the cost and size of the converter. This progress is achieved by adding four low-frequency switches to the conventional configuration of SM converter whereas the number and voltage rating of high-frequency switches and clamping capacitors as well as the number of high-frequency switchings during a full cycle are kept constant. This converter is controlled by phase shifted carrier-sinusoidal pulse width modulation technique; therefore the self-balancing phenomenon of clamping capacitors' voltages is maintained. This study also presents a state-space representation model to analyse the dynamic of clamping capacitor's self-balancing phenomenon in the proposed SM converter by obtaining the switching instants of the pulse width modulation in terms of the Kapteyn series. Numerical solution of obtained state-space representation model of the proposed converter and simulation results as well as measurements taken from an experimental set-up are presented in order to validate the effectiveness and advantages of the proposed configuration as well as its control strategy and state-space model.

Methods for summing general Kapteyn series

The general features and characteristics of Kapteyn series, which are a special type of series involving the Bessel function, are investigated. For many applications in physics, astrophysics and mathematics, it is crucial to have closed-form expressions in order to determine their functional structure and parametric behavior. The closed-form expressions of Kapteyn series have mostly been limited to special cases, even though there are often similarities in the approaches used to reduce the series to analytically tractable forms. The goal of this paper is to review the previous work in the area and to show that Kapteyn series can be expressed as trigonometric or gamma function series, which can be evaluated in a closed form for specific parameters. Two examples with a similar structure are given, showing the complexity of Kapteyn series.

Integral representation of first kind Kapteyn series

The main aim of this research note is to establish two different types of integral representation formulae for the Kapteyn series of the first kind. The first one is a double definite integral expression, while the second type includes indefinite integral representation formula.

On Taylor series and Kapteyn series of the first and second type

We study the relation between the coefficients of Taylor series and Kapteyn series representing the same function. We compute explicit formulas for expressing one in terms of the other and give examples to illustrate our method.

On an integral representation of a class of Kapteyn (Fourier–Bessel) series: Kepler’s equation, radiation problems and Meissel’s expansion

In this paper, an integral representation of a class of Kapteyn series is proposed. Such a representation includes the most used series in practical applications. The approach uses the property of uniform convergence of the considered class and the integral representation of the Bessel functions. The usefulness of the proposed method is highlighted by providing an integral solution of Kepler’s equation and of some Kapteyn series arising in radiation problems. Moreover it allows us to generalize a result due to Meissel.

Spectral Analysis of a Class of DC–AC PWM Inverters by Kapteyn Series

The pulsewidth modulation (PWM) technique for single-phase bipolar switching inverters has been revisited and analyzed on the basis of the isomorphism between the implicit equation of the unknown PWM switching instants, when the input to the modulator is a sinusoidal signal of period multiple of the carrier period, and the Kepler's equation governing the trajectory of a planet around the sun. The solution of the Kepler's problem allows to obtain the switching instants of the PWM in terms of Kapteyn series or integral forms and to find an explicit expression of the harmonic content. An expression of the weighted total harmonic distortion is also obtained in terms of Bernoulli polynomials.

Kapteyn series in high intensity Compton scattering

In discussing the signatures available from high intensity Compton scattering, a problem of considerable interest in photon scattering from active galactic nuclei and their emission jets, Harvey et al (2009 Phys. Rev. A 79 063407) showed that the properties of some other than conventional Kapteyn series play fundamental roles in determining the spectral output both with respect to frequency and with respect to emission angle. While they were able to provide bounds to the required series in terms of known Kapteyn series that could be summed analytically in closed form, only numerical analysis could take their investigation further as they demonstrated. The purpose of this paper is to show that the many Kapteyn series involved in the scattering problem can all be reduced either to analytic form or to a single Kapteyn series that cannot be evaluated in closed form but for which an integral representation is available. This reduction is of considerable benefit in controlling the correctness and accuracy of numerical investigations; the reduction also provides significant insight into some basic procedures for summing such unconventional Kapteyn series as well as allowing a better understanding of the dependence of the scattering on the physical parameters involved than would be the case directly from the series.

The analytical summation of a new class of Kapteyn series

Exact expressions for certain Kapteyn series of the first and second kind are obtained. Such sums are needed in analytical theories describing Cosmic Ray scattering in turbulent magnetic fields. Also, some functions defined by sums involving derivatives of Bessel functions are considered. Numerical methods and applications of the results are discussed.

Terahertz-sideband spectra involving Kapteyn series

Kapteyn series of the second kind 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 (Citrin D S 1999 Phys. Rev. B 60 5659) and in certain time-periodic transport problems in superlattices (Ignatov A A and Romanov Y A 1976 Phys. Status Solidi b 73 327; Feise M W and Citrin D S 1999 Appl. Phys. Lett. 75 3536). This paper 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. The general method of analysis may also be of interest for other Kapteyn series.

A Review of Procedures for Summing Kapteyn Series in Mathematical Physics

Since the discussion of Kapteyn series occurrences in astronomical problems the wealth of mathematical physics problems in which such series play dominant roles has burgeoned massively. One of the major concerns is the ability to sum such series in closed form so that one can better understand the structural and functional behavior of the basic physics problems. The purpose of this review article is to present some of the recent methods for providing such series in closed form with applications to: (i) the summation of Kapteyn series for radiation from pulsars; (ii) the summation of other Kapteyn series in radiation problems; (iii) Kapteyn series arising in terahertz sideband spectra of quantum systems modulated by an alternating electromagnetic field; and (iv) some plasma problems involving sums of Bessel functions and their closed form summation using variations of the techniques developed for Kapteyn series. In addition, a short review is given of some other Kapteyn series to illustrate the ongoing deep interest and involvement of scientists in such problems and to provide further techniques for attempting to sum divers Kapteyn series.