Summation Of Series Research Articles

Feynman Perturbation Series for the Morse Potential

In this paper we give an alternative treatment of the Schrodinger equation with the Morse potential, which based on the exact summation of the Feynman perturbation series in its original form. Using Fourier transform we establish a recurrence equation between terms of the perturbation series. Finally, by the inverse Fourier transform and some technical tools of the ordinary differential equations of the second order, we can compute the exact sum of the perturbation series which is the Green’s function of the problem.

Power series with binomial sums and asymptotic expansions

This paper is a study of power series, where the coefficients are binomial expressions (iterated finite differences). Our results can be used for series summation, for series transformation, or for asymptotic expansions involving Stirling numbers of the second kind. In certain cases we obtain asymptotic expansions involving Bernoulli polynomials, poly-Bernoulli polynomials, or Euler polynomials. We also discuss connections to Euler series transformations and other series transformation formulas. Mathematics Subject Classification: Primary 11B83, Secondary 33B15, 40C15, 05A15

Energy Detection in Multihop Cooperative Diversity Networks: An Analytical Study

This work presents a study of detection performance of energy detector in relay-based multihop cooperative diversity networks operating over independent Rayleigh fading channels. In particular, upper bound average detection probability expressions are obtained for three scenarios: (i) multihop cooperative relay communication; (ii) multi-hop cooperative relay communication with direct link and maximum ratio combiner (MRC) at destination; and (iii) multi-hop cooperative relay communication with direct link and selection combiner (SC) at destination. Classical method of performance evaluation based on probability density function (PDF) of received signal-to-noise ratio (SNR) is used. Furthermore, an alternative series form representation of generalized Marcum- Q function is employed in order to simplify the ensuing mathematical derivations. Because the obtained detection probability expressions are in the form of infinite summation series, their respective truncation error bounds are also derived. In the end, analyses are validated with the help of simulations and several observations regarding the impact of different parameters on detector's performance are outlined.

On the interactions of the high energy photoelectrons with the fullerene shell

The probability that photoionization of the caged atom in an endohedral system is accompanied by excitation of the fullerene shell is shown to be close to unity in broad intervals of the photoelectron energies. The result is obtained by summation of the perturbative series for the interaction between the photoelectron and the fullerene shell. This result means that the interaction between the photoelectron ejected from the cage atom and the fullerene shell cannot be described by a static potential, since inelastic processes become decisively important.

The low frequency axisymmetric modal sound radiation efficiency of an elastically supported annular plate

Abstract This paper deals with the sound radiation efficiency of a vibrating, thin, elastically supported annular plate embedded into a flat rigid baffle. The free axisymmetric time harmonic vibrations have been considered for a single mode. It has been assumed that the influence of the air column above the plate on the plate's vibrations is negligible. First, the sound radiation efficiency has been formulated as an integral. Further, rigorous mathematical manipulations have been carried out based on the theory of summation of multiple expansion series containing the hypergeometric functions. As a result, the formulations have been expressed as some fast convergent expansion series containing only the Bessel and Struve functions of integer order and the spherical Bessel functions. The presented formulations of sound radiation efficiency of an elastically supported annular plate are useful for numerical calculations within the low frequency range what is important for practical reasons. The formulations are valid for axisymmetric boundary conditions and they enable changing the values of boundary stiffness constants. Consequently, the analysis of influence of the plate's edge attachment on the sound radiation efficiency has been performed. The limiting transitions have also been performed from formulations valid for the elastically supported annular plates to the formulations valid for annular plates with classical boundary conditions (clamped, simply supported and free) at one edge or at both edges.

Green Function of Quantum Liouville Problem via Exact Summation of Perturbation Series

Liouville Problem | Perturbation Series | Fourier Transform | Green Function | Problème de Liouville | Séries de Perturbation | Transformation de Fourier | Fonction de Green | مسألة ليوفيل | سلاسل الاضطراب | تحويل فوري | دالة غرين

Numerical Evaluations of Diffraction Formulas for the Canonical Wedge-Scattering Problem [Testing Ourselves

Numerical computations of different diffraction representations for the canonical wedge-scattering problem are discussed. Representations include total-field exact series summation, exact diffraction integral, and the Physical Optics (PO), Physical Theory of Diffraction (PTD), and Parabolic Equation (PE) techniques, and the Uniform Theory of Diffraction (UTD). Numerical challenges and advantages of each are given.

New Recursive Representations for the Favard Constants with Application to the Summation of Series

New Recursive Representations for the Favard Constants with Application to the Summation of Series

Pair distribution function analysis of X-ray diffraction from amorphous spheres in an asymmetric transmission geometry: application to a Zr58.5Cu15.6Ni12.8Al10.3Nb2.8glass

A method for the calculation of the pair distribution and structure functions from X-ray intensity data obtained with an area detector for an off-center incident X-ray beam on an amorphous sphere is presented. Error propagation for converting from the structure function to the pair distribution function is also described, including a summation series approach to treat the error from a high-qtruncation. A Zr58.5Cu15.6Ni12.8Al10.3Nb2.8glass (Vitreloy 106a) is used to demonstrate the techniques. In particular, the semi-analytical corrections presented to calculate the effects of secondary scatter within and asymmetric transmission through a spherical sample are verified.

Extending the rubber rope: convergent series, divergent series and the integrating factor

A well-known mathematical puzzle regarding a worm crawling along an elastic rope is considered. The resulting generalizations provide examples for use in a teaching context including applications of series summation, the use of the integrating factor for the solution of differential equations, and the evaluation of definite integrals. A number of potential classroom exercises are given.

Tight upper bounds on average detection probability in cooperative relay networks with selection combiner

AbstractIn this study, the detection probability for amplify–forward‐based cooperative relay networks, where the destination is equipped with a selection combiner and energy detector, is analysed. The probability density function method and an alternative series representation of the generalized Marcum‐Q function are used to obtain two upper bounds for the average detection probability. The error bounds resulting from truncating an infinite summation series to finite terms are also calculated because the derived expressions were in the form of an infinite summation series. Finally, the analysis are validated by simulations. These results are expected to assist in future studies of communication technologies, such as cognitive radios. Copyright © 2013 John Wiley & Sons, Ltd.

Fluctuating parts of nuclear ground-state correlation energies

Background: Heavy atomic nuclei are often described using the Hartree-Fock-Bogoliubov (HFB) method. In principle, this approach takes into account Pauli effects and pairing correlations while other correlation effects are mimicked through the use of effective density-dependent interactions. Purpose: Investigate the influence of higher order correlation effects on nuclear binding energies using Skyrme's effective interaction. Methods: A cut-off in relative momenta is introduced in order to remove ultraviolet divergences caused by the zero-range character of the interaction. Corrections to binding energies are then calculated using the quasiparticle-random-phase approximation (QRPA) and second order many-body perturbation theory (MBPT2). Result: Contributions to the correlation energies are evaluated for several isotopic chains and an attempt is made to disentangle which parts give rise to fluctuations that may be difficult to incorporate on the HFB level. The dependence of the results on the cut-off is also investigated. Conclusions: The improved interaction allows explicit summations of perturbation series which is useful for the description of some nuclear observables. However, refits of the interaction parameters are needed to obtain more quantitative results.

Diffraction of acoustic and elastic waves on a half-plane for boundary conditions of various types

The classical problem of wave diffraction on a half-plane with boundary conditions of different types and its generalizations to elastic media are considered. As a solution method it is proposed to combine the Fourier method of separation of variables and the series summation technique based on the use of integral representations of Bessel functions. The analytic solutions thus obtained are equally efficient in the near- and far-field diffraction regions. The two-term singularity at a corner point (in stresses for elastic media and in the velocity for acoustic media) was discovered for the first time. The knowledge of singularities in the scalar problem allowed one to construct the solution of the vector problem of elastic longitudinal wave diffraction. It is investigated how different types of boundary conditions on both sides of the half-plane affect the solution behavior in the far-field region. Possible physical interpretations of the obtained results are given.

Multi-scale analysis of linear data in a two-dimensional space

Many disciplines are faced with the problem of handling time-series data. This study introduces an innovative visual representation for time series, namely the continuous triangular model. In the continuous triangular model, all subintervals of a time series can be represented in a two-dimensional continuous field, where every point represents a subinterval of the time series, and the value at the point is derived through a certain function (e.g. average or summation) of the time series within the subinterval. The continuous triangular model thus provides an explicit overview of time series at all different scales. In addition to time series, the continuous triangular model can be applied to a broader sense of linear data, such as traffic along a road. This study shows how the continuous triangular model can facilitate the visual analysis of different types of linear data. We also show how the coordinate interval space in the continuous triangular model can support the analysis of multiple time series through spatial analysis methods, including map algebra and cartographic modelling. Real-world datasets and scenarios are employed to demonstrate the usefulness of this approach.

On “dynamical mass” generation in Euclidean de Sitter space

We consider the perturbative treatment of the minimally coupled, massless, self-interacting scalar field in Euclidean de Sitter space. Generalizing work of Rajaraman, we obtain the dynamical mass of the scalar for nonvanishing Lagrangian masses and the first perturbative quantum correction in the massless case. We develop the rules of a systematic perturbative expansion, which treats the zero-mode nonperturbatively and goes in powers of $\sqrt{\ensuremath{\lambda}}$. The infrared divergences are self-regulated by the zero-mode dynamics. Thus, in Euclidean de Sitter space the interacting, massless scalar field is just as well defined as the massive field. We then show that its dynamical mass ${m}^{2}\ensuremath{\propto}\sqrt{\ensuremath{\lambda}}{H}^{2}$ can be recovered from the diagrammatic expansion of the self-energy and a consistent solution of the Schwinger-Dyson equation, but it requires the summation of a divergent series of loop diagrams of arbitrarily high order. Finally, we note that the value of the long-wavelength mode two-point function in Euclidean de Sitter space agrees at leading order with the stochastic treatment in Lorentzian de Sitter space, in any number of dimensions.

Efficient and Accurate Approximation of Infinite Series Summation Using Asymptotic Approximation and Super Convergent Series

We present an approach for very quick and accurate approximation of infinite series summation arising in electromagnetic problems. This approach is based on using asymptotic expansions of the arguments and the use of super convergent series to accelerate the convergence of each term. It has been validated by obtaining very accurate solution for propagation constant for shielded microstrip lines using spectral domain approach (SDA). In the spectral domain analysis of shielded microstrip lines, the elements of the Galerkin matrix are summations of infinite series of product of Bessel functions and Green's function. The infinite summation is accelerated by leading term extraction using asymptotic expansions for the Bessel function and the Green's function and the summation of the leading terms is carried out using the super convergent series.

New Recursive Representations for the Favard Constants with Application to Multiple Singular Integrals and Summation of Series

There are obtained integral form and recurrence representations for some Fourier series and connected with them Favard constants. The method is based on preliminary integration of Fourier series which permits to establish general recursion formulas for Favard constants. This gives the opportunity for effective summation of infinite series and calculation of some classes of multiple singular integrals by the Favard constants.

Summation of Multiple Fourier Series in Matrix Weighted -Spaces

This paper is concerned with rectangular summation of multiple Fourier series in matrix weighted -spaces. We introduce a product Muckenhoupt condition for matrix weights and prove that rectangular Fourier partial sums converge in the corresponding matrix weighted space , , if and only if the weight satisfies the product Muckenhoupt condition. The same result is shown to hold true for other summation methods such as Cesàro and summation with the Jackson kernel.

On a treatment of coupled integral equations in nonlinear unilateral contact problem of plates

Application of finite Hankel integral transforms to the problem of a uniformly loaded unilaterally supported rectangular thin plates, the solution of partial differential equation can then be reduced to two coupled Fredholm integral equations of the second kind that obtained from the coupled dual-series equations of mixed boundary conditions, and also together with two constraint integral equations resulted from both of zero corner force condition and compatibility of edge displacements near each plate corner that loses contact. Owing to the summation of many different infinite series in the kernels and inhomogeneous parts of equations for the requirements of attainable highest accuracy and reducible computational time, the modified mathematical manipulations are improved and required asymptotic expansion for large arguments of the Bessel and Struve functions is treated to accelerate the convergence of series rapidly. The latter is to avoid the undesirable lengthy computer runs in numerical computations. Consequently, this paper is devoted to demonstrate the mathematical description and numerical algorithms in order to solve the system of coupled integral equations.

An experimental conjecture involving closed-form evaluation of series associated with the Zeta functions

The subject of closed-form summation of series involving the Zeta functions has been remarkably widely investigated. Recently, in the course of his trying to give a closed-form expression for the Dirichlet beta function β(2n )( n ∈ N) ,L ima (16) posed a very interesting experimental conjecture for a closed-form evaluation of a certain class of series involving the Riemann Zeta function ζ(s). Here, in the present sequel to Lima's work, we aim at verifying correctness of Lima's conjecture and presenting several general analogues of Lima's conjecture. Our demonstration and derivations are based mainly upon a known formula for series associated with the Zeta functions. Relevant connections of some specialized results of the main identities presented here with those obtained in earlier works are also pointed out.