IV. On infinite series.

Some applications of fractional calculus involving summation of infinite series

(1907). Summation of Certain Infinite Series. The American Mathematical Monthly: Vol. 14, No. 3, pp. 41-46.

In Sect. 1.9 we considered summation of finite series. In practice it is often necessary to deal with infinite series which contain an infinite number of terms a n . Here a n is a general (we say “the n-th”) term of the series which is constructed via a some kind of formula depending on an integer n = 1, 2, 3, …. For instance, the rule \(a_{n} = a_{0}q^{n}\) with n = 0, 1, 2, 3, … corresponds to an infinite geometric progression $$\displaystyle{a_{0} + a_{0}q + a_{0}q^{2} + \cdots = a_{ 0}\left (1 + q + q^{2} + \cdots \,\right ).}$$

Summation of Certain Infinite Series

In this paper we study the cohomological Conley index of arbitrary isolated invariant continua for continuous maps $$f :U \subseteq {\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$$ by analyzing the topological structure of their unstable manifold. We provide a simple dynamical interpretation for the first cohomological Conley index, describing it completely, and relate it to the cohomological Conley index in higher degrees. A number of consequences are derived, including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3. Our approach exploits a certain attractor–repeller decomposition of the unstable manifold, reducing the study of the cohomological Conley index to the relation between the cohomology of an attractor and its basin of attraction. This is a classical problem that, in the present case, is particularly difficult because the dynamics is discrete and the topology of the unstable manifold can be very complicated. To address it we develop a new method that may be of independent interest and involves the summation of power series in cohomology: if Z is a metric space and $$K \subseteq Z$$ is a compact, global attractor for a continuous map $$g :Z \rightarrow Z$$, we show how to interpret series of the form $$\sum _{j \ge 0} a_j (g^*)^j$$ as endomorphisms of the cohomology group of the pair (Z, K).

The theory of distributions has a flavor similar to the theory of summation of divergent series and integrals and, as we have seen in Chapter 6, is closely related to the theory of singular integrals. A generic alternating series $$1 - 1 + 1 - 1 + \ldots = \sum { \pm 1} $$ is a good example here. It seems to make no sense to assign a specific value to this infinite sum. Nevertheless, mathematicians have produced certain reasonable rules of summation that assign to it value 1/2. Such an assignment is in complete agreement with the intuition of physicists who encounter similar series. In this chapter, we will see how one can sum this, or even more strange, divergent series and integrals. To gain a better insight into the essence of this problem, let us begin with elementary examples and recall basic notions and theorems of the ordinary theory of convergent infinite series.

The problem of summation of Fourier series in finite form is formulated in the weak sense, which allows one to consider this problem uniformly both for classically convergent and for divergent series. For series with polynomial Fourier coefficients \(a_n, b_n \in \mathbb{R}[n]\), it is proved that the sum of a Fourier series can be represented as a linear combination of 1, \(\delta(x)\), \(\cot \frac{x}{2}\) and their derivatives. It is shown that this representation can be found in a finite number of steps. For series with rational Fourier coefficients \(a_n, b_n \in \mathbb{R}(n)\), it is shown that the sum of such a series is always a solution of a linear differential equation with constant coefficients whose right-hand side is a linear combination of 1, \(\delta(x)\), \(\cot \frac{x}{2}\) and their derivatives. Thus, the issue of summing a Fourier series with rational coefficients is reduced to the classical problem of the theory of integration in elementary functions.

By a series we mean a set of numbers a1, a2, a3… such that we have a rule for calculating a2, a3 etc. from the first number a1.Series occur in many problems in chemistry such as specific heats of solids, the theory of black-body radiation, solution of the Schrödinger equation, statistical thermodynamics and Fourier series in X-ray crystallography. In this chapter we consider the summation of series and the convergence of infinite series. We shall develop methods for the expansion of functions such as transcendental functions in terms of infinite series.

We first consider a summation procedure for some trigonometric series in terms of the Riemann zeta and related functions. In some cases these series can be brought in closed form, which means that the infinite series are represented by finite sums. Afterwards, we show some applications of our results to the summation of series involving Bessel or Struve functions. Further, relying on results from the previous sections, we obtain sums of series involving a Bessel or Struve integral. These series are also represented as series in terms of the Riemann zeta and related functions of reciprocal powers and can be brought in closed form in certain cases as well. By replacing the function appearing in a Bessel and Struve integral with particular functions, we find sums of new series.

In learning complex analysis, the residue theorem is the easiest thing to touch. Then in higher mathematics, these concepts are often very abstract, and all exist in graphics. Therefore, the author summarizes the practical application of the residue theorem with reference to other papers. There are many application directions. The simplest and practical application is in generalized integral. This paper introduces the research study topic of the application of generalized residue theorem in Laplace Inverse transform. The application methods and skills of generalized residue theorem in Laplace Inverse transform are discussed in detail through concrete examples. Based on the teaching process of Laplace transformation in mathematical analysis, differential equation or mathematical equation, and the problems and puzzles in the application of Laplace transformation, this paper presents a research-based learning theme. In this paper on the application of residue theorem to series, a new method of summation of infinite series is given based on residue theorem. In this method, the sum of series is transformed into the sum of residues of a complex valued function in a closed domain.

This article presents new constructive formulas to steady-state thermoelastic Green’s functions for a plane generalized boundary value problem of thermoelasticity for a generalized rectangle. The constructive formulas are expressed in terms of Green’s functions for Poisson’s equation. These results are formulated in a special theorem, which is proved using the author’s developed harmonic integral representations method. On the base of derived constructive formulas, it is possible to obtain many analytical expressions for Green’s functions for thermoelastic displacements and stresses to 28 concrete boundary value problems for: rectangle-16, half-strip-8, strip-4. An example of such kind is presented for a concrete plane boundary value problem for a rectangle, Green’s functions of which are presented in the form of a sum of elementary functions and ordinary series. These results are presented in another theorem, which is proved on the base of derived general constructive formulas. In the particular case for a half-strip and strip, ordinary series vanish and Green’s functions are presented in elementary functions. New analytical expressions for thermal stresses to a particular plane problem for a thermoelastic rectangle subjected to a constant boundary temperature gradient are also derived. Numerical investigation has shown that the infinite series are convergent. All solutions obtained for thermal stresses, caused by a constant temperature gradient and by a unit heat source are validated by checking the respective equilibrium equation and continuity of deformation equations (Beltrami–Michel equations), written in the terms of thermal stresses. The graphics for thermal stresses and their infinite series also are presented.

We prove the basic trigonometric Korovkin approximation theorem for fuzzy valued functions of two variables and verify the approximation by the help of fuzzy modulus of continuity. Also, we introduce double level Fourier series of fuzzy valued functions and investigate corresponding approximation through the use of Ces\`{a}ro and Abel methods of summation of infinite series.

The linear Boltzmann equation for the transport of neutral particles is investigated with the objective of generating a benchmark-quality calculation for the three-dimensional searchlight problem in a semi-infinite medium. The derivation assumes stationarity, one energy group, and isotropic scattering. The scalar flux (both surface and interior) and the current at the surface are the quantities of interest. The source considered is a pencil-beam incident at a point on the surface of a semi-infinite medium. The scalar flux will have two-dimensional variation only if the beam is normal; otherwise, it is three-dimensional. The solutions are obtained by using Fourier and Laplace transform methods. The transformed transport equation is formulated so that it can be related to a one-dimensional pseudo problem, thus providing some analytical leverage for the inversions. The numerical inversions use standard numerical techniques such as Gauss-Legendre quadrature, summation of infinite series, H-function iteration and evaluation, and Euler-Knopp acceleration. The numerical evaluations of the scalar flux and current at the surface are relatively simple, and the interior scalar flux is relatively difficult to calculate because of the embedded two-dimensional Fourier transform inversion, Laplace transform inversion, and H-function evaluation. Comparisons of these numerical solutions to results from the MCNP probabilistic code and the THREEDANT discrete ordinates code are provided and help confirm proper operation of the analytical code.

A theoretical analysis of the predicted performance and field distributions for semiconductor junction circulators is based on Bosma's Green's function approach which involves a summation of infinite series in the mathematical derivation. The effects are considered of taking a different number of terms in the series, looking at three different circulators centred at 120 GHz. The narrowband semiconductor circulators show a similar effect to the ferrite junction circulator because the field distributions inside the semiconductor discs can be approximated by only considering the dominant modes. However, there is no dominant mode in the broadband circulator and the higher order modes play an important role in the operation of this circulator. In order to obtain a precise representation of the circulation effects inside both narrowband and broadband circulators, at least up to the 6th term, or 18 terms in all, are required to be added in the summation of Green's functions.

It was over 2,000 years ago that Zeno propounded his now famous arguments against motion. The four paradoxes he suggests allegedly lead to the conclusion that motion is impossible, and have sparked comments by philosophers as diverse as Aristotle, Hegel, and Russell. In the present paper, I will consider Zeno's first paradox—the Dichotomy--and the modern attempt to discount it. I will contend that the most philosophically interesting version of the Dichotomy undermines the widely held claim that the dilemma it presents can be resolved by an appeal to mathematical truths about the summation of an infinite series. If this is the case, then philosophers like Robinson, 1 Burnet, 2 Booth, 1 Russell, Whitehead,* Carnap,* Quine, 7 Vlastos,' and others have failed to recognize the fundamental questions the Dichotomy raises. Nevertheless, their account of the paradox is the predominant one,' and has been expounded in books by Grunbaum 1 0 and Salmon. 1 1 Indeed, it has been claimed (by Maxwell and Feigl) 1 2 that the summation answer to Zeno's reasoning provides a paradigm solution to a philosophical problem, and (by Boyer) 1 1 that it demonstrates the power of the calculus. Within contemporary philosophy, it has also sparked the discussion of 'infinity machines' by Grunbaum, Vlastos, Black, l k Thomson, 1 1 Benacceraf, 1 4 and others. The final outcome of this discussion has been the account of infinite sequences of acts adopted by philosophers like Grunbaum, Vlastos, Putnam, 1 7 and Boolos and Jeffrey.' If the position I argue for is correct however, then all of these philosophers--and philosophers (like C. D. Broad* and Philip Jones) '• who apply similar considerations to Zeno's Achilles paradox—have failed to come to grips with the fundamental issues raised by Zeno's Dichotomy argument.