Summation Of Infinite Series Research Articles

A very simple computation of the Casimir effect

There are several (in the end equivalent) methods for obtaining the vacuum energy density corresponding to the Casimir effect. Here, a detailed, pedagogical account is provided of the most elegant of them, namely the zeta-function regularization procedure. It is shown how it yields (in many instances)exact results, without ever having to resort to the usual issue of infinity cancellations. This extremely simple way of dealing—by analytic continuation—with the series of eigenmodes, can be made into a closed procedure owing to a recent theorem, which extends a previous result of Weldon on the commutation of the order of the summations of infinite series. The cases of a scalar field with Dirichlet, Neumann and periodic boundary conditions, and of an electromagnetic field between perfectly conducting plates—intimately related from a mathematical point of view—are investigated.

A high order renormalization method for radar backscatter from a random medium

The problem of radar backscattering from a half-space random medium is formulated in terms of a series solution to Dyson's equation for the mean field in the random medium. The renormalization method is applied to given an approximate solution through the use of Feynman diagrams. An improved approximation is then obtained by including two additional summations of infinite series of these diagrams. The results from this improved approximation are used to calculate the radar backscattering cross sections and compared with those of a previous paper (1980). For comparison with experimental data, a field of vegetation is modeled as a random medium with an average complex permittivity as well as fluctuations in the real and imaginary parts of the permittivity. The values of the average permittivity and the variance of the fluctuations are estimated by the use of dielectric mixture theory. >

Green's functions and the diagrammatic technique in the theory of gas-phase electron diffraction on vibrationally excited polyatomic molecules. Part I.

A new method for calculating thermodynamical averages of arbitrary functions of normal coordinates for polyatomic molecules is proposed, based on the calculation of the multi-point thermodynamic Green's functions. The general theory is applied to evaluation of the molecular scattering function for gas-phase electron diffraction. It is shown that the present theory enables a practical summation of infinite perturbative series to account for an anharmonicity in vibrations, a correct description of the Fermi resonances to all orders of perturbation theory, and it is possible to perform the calculations for highly excited vibrational states of polyatomic molecules, without evaluation of the molecular wavefunctions.

Dissipation of the partial ground fault current across the shield wires of transmission lines

An analytical method is presented which can be used to estimate grounding effects of the shield wires, voltage transfer and distribution of ground fault currents. In the case of lines which can, as regards earthing, be treated as infinite, the method is based on relatively simple equivalent circuits obtained by applying the expression for the earthing impedance of an infinite line and summation of finite geometric series. In the case of lines which cannot be treated as infinite, use is also made of the principle of superposition and the summation of infinite geometric series. Some numerical examples are given to illustrate the practical application of the method. Finally, the method is used to derive an analytical proof of the specific property of shield wires as earthing conductors that they partially compensate for the unfavorable influence of high soil resistivity on the resistance of the earthing electrodes which they connect.

Prediction of long-term leachability of a solidified radioactive waste from a short-term leachability test by a similitude law for leaching systems

A Similitude Law for Leaching Systems has been developed based on a one-dimensional mass transport theory which includes the processes of diffusion, sorption, and radioactive decay. The similitude law can be used to determine the leaching characteristics of a solidified waste from a laboratory leaching test conducted for a similar test piece and to determine the long-term leaching characteristics of a test piece from the results of a short-term test. A leaching number developed from the similitude law was found to be the key parameter in characterizing leach test systems. A mathematical model based on one-dimensional transport theory was also developed to determine the general leaching characteristics of a solidified low-level radioactive waste in a test leachant. The results were normalized, based on the simililtude law, in a leaching graph which uses the leaching number as a system parameter. The leaching number can be obtained through short-term leaching tests. The normalized leaching graph can be used to determine the leaching characteristics of a solidified waste without undergoing computer simulation or tedious numerical summations of infinite series.

Integration by parts: The algorithm to calculate β-functions in 4 loops

The following statement is proved: the counterterm for an arbitrary 4-loop Feynman diagram in an arbitrary model is calculable within the minimal subtraction scheme in terms of rational numbers and the Riemann ζ-function in a finite number of steps via a systematic “algebraic” procedure involving neither integration of elementary, special, or any other functions, nor expansions in and summation of infinite series of any kind. The number of steps is a rapidly increasing function of the complexity of the diagram. To demonstrate further possibilities offered by the technique we compute the 5-loop diagram contributing to the anomalous field dimension γ 2( g) in the ϕ 4 model, that defied, heretofore, analytical calculation by other methods.

Path integrals in chemical kinetics II

Proceeding from the approach by Zel'dovich and Ovchinnikov, we develop an approximate method for the solution of master equations, based on the summation of infinite diagrammatic series of perturbation theory. As an illustration, an example of a diffusion-controlled chemical reaction is analysed.

Calculation of the transport coefficients for a dilute simple gas using an iterative solution of the Boltzmann equation

The Chapman-Enskog second approximation to Boltzmann's equation for a dilute simple gas is obtained for several different potential models using an iterative procedure. The perturbation functions to the local Maxwellian distribution, and various transport coefficients, are found as approximate sums of infinite series. Transport coefficients are calculated for Maxwellian, pseudo-Maxwellian, and rigid-sphere potential models and are compared with those obtained by the standard Chapman-Cowling method.

Analysis of excitation energies and transition moments

The lowest electronic transitions in H2, CH+ and Be have been analysed in terms of the individual contributions to excitation energies and transition moments in a second-order polarisation propagator calculation and in a second-order diagrammatic calculation. These are based on both the full particle-hole propagator matrix and on the single-transition approximation. Collective effects, defined as sums of infinite series of irreducible self-energy diagrams, do not affect the excitation energies but change the intensities by up to 40 per cent. The cancellation effects among the second-order contributions to the excitation energy are substantial, indicating the importance of using methods using methods which are consistent through a certain order in the electronic repulsion.

Partial Sums of Infinite Series, and How They Grow

Partial Sums of Infinite Series, and How They Grow

Partial Sums of Infinite Series, and How They Grow

(1977). Partial Sums of Infinite Series, and How They Grow. The American Mathematical Monthly: Vol. 84, No. 4, pp. 237-258.

Some combinatorial identities of Sarmanov, Sevast'yanov, and Tarakanov

Sarmanov, Sevast'yanov, and Tarakanov have proved certain combinatorial identities by a combinatorial argument, namely, by counting the number of weighted chains by two different methods. The identities are applied to the summation of infinite series and to the computation of the quartiles of discrete probability distributions. Egorychev and Yuzhakov have proved these and some further identities by applying the multidimensional generalization of the Lagrange expansion. In the present paper the identities are proved in a direct, elementary way.

Stress-Focusing Effect in a Uniformly Heated Cylindrical Rod

A long cylindrical rod is considered brought suddenly to a uniform temperature rise over its cross section. Stress-focusing effects occur when stress waves reflect from the outer surface of the rod and proceed radially inward to the axis. The focusing effect can cause a very high peak dynamic stress in both tension and compression in the rod. The magnitude of the peak stress depends upon the magnitude of the temperature rise and the effective heating duration. For instantaneous heating, the infinite peak of stress propagates outward from the center while these peaks are finite for nonzero heating duration. The solutions are carried out by using Laplace transform on time and presented as infinite series summations after the end of heating.

Methods for the Summation of Infinite Series in Closed Form

The paper discusses four analytical methods for the summation of infinite series in closed form. Many illustrative examples are given including some with direct relevance to electrical engineering subjects like Fourier analysis of circuits, potential solutions of Laplace's equation, and finite differences.

Some series related to infinite series given by Ramanujan

In this paper we discuss some formulas concerning the summation of certain infinite series, given by Ramanujan in his notebooks [1], vol. 1, Ch. XVI (pp. 251–263), and vol. 2, Ch. XV (pp. 181–192). (A large part of the material in Ch. XVI is contained also in Ch. XV, with only minor changes.) It is shown that several of the formulas given are erroneous. Most of the remaining formulas have by now been proved by residuum calculus. Some of these proofs are extendable to cases which do not seem to have attracted attention earlier. As an example of this we mention the sums $$\sum\limits_{n = 1}^\infty {( - 1)^{n - 1} n^s } /\sinh n\pi = 0 for s = 5,9,13,17,...$$

Some combinatorial identities

The paper introduces a series of new combinatorial identities which allow one to find the value of the partial sums of some multiple power series. Examples are given of the application of these identities to the summation of infinite series and to the computation of the quantiles of discrete probability distributions.

On Absolute Summability by Riesz and Generalized Cesàro Means. I

1. The Cesàro methods for ordinary [9, p. 17; 6, p. 96] and for absolute [9, p. 25] summation of infinite series can be generalized by the Riesz methods [7, p. 21; 12; 9, p. 52; 6, p. 86; 5, p. 2] and by “the generalized Cesàro methods“ introduced by Burkill [4] and Borwein and Russell [3]. (Also cf. [2]; for another generalization, see [8].) These generalizations raise the question as to their equivalence.We shall consider series(1)with complex terms an. Throughout, we will assume that(2)and we call (1) Riesz summable to a sum s relative to the type λ = (λn) and to the order κ, or summable (R, λ, κ) to s briefly, if the Riesz means(of the partial sums of (1)) tend to s as x → ∞.

The Backward and Forward Summation of Infinite Series of Isols.

The Backward and Forward Summation of Infinite Series of Isols.

On the summation of certain legendre series

The purpose of this paper is to find some recurrence relations for sums of infinite series of the formσ P l n (cosθ)tl+m/(l+m). This is achieved by transforming the sums into integrals and then using the recurrence relations for these integrals.

Turbulent heat transfer in a tube with prescribed heat flux

An approximate method is proposed to predict Nusselt number for turbulent flow in a tube with an arbitrary axially varying heat flux distribution. Evaluation of sums of infinite series with poor convergency is required for this method and done correctly. The predicted results by the proposed method were examined experimentally with air for uniform heat flux, exponentially varying heat flux and rectilinearly varying heat flux. The experimental apparatus was so constructed that heat flux could distribute in the prescribed manner by changing axially the cross sectional area of the tube wall conducted low voltage a.c. Predicted Nusselt numbers are in good agreement with the experimental data. The conditions for heat flux distributions which have the fully developed situation with generalized temperature profile invariant in the axial direction were analysed theoretically. As the result, it is found the exponentially varying heat flux alone can realize the fully developed situation. Both cases of uniform heat flux and uniform wall temperature, which have been well known to realize the fully developed situation, correspond to the special cases of exponentially varying heat flux.