Infinite Numerical and Functional Series

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

I. In the Paper, which the Royal Society did me the honour to print, on Summation of Series, is given a method of finding the sum of series

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).

In this work we apply the classical Plana summation formula to derive integral representations for certain infinite series that their general terms containing Gamma, q q -Gamma, theta, Bessel, and Hurwitz zeta functions.

In the author's paper [7] it was proved that the fractional parts of the partial sums of an infinite series (of real terms) diverging to +∞ or −∞, in which the general term tends to zero, are everywhere dense in the closed unit interval. This result was extended to series of infinite oscillation (see Remark 4.1 of the said paper) on the argument that a sequence of partial sums having infinite oscillation has a subsequence that diverges to +∞ or −∞.

Some applications of fractional calculus involving summation of infinite series

A unified framework is presented for nonlinear (and, in particular, linear) system and signal analysis, whereby a number of problems involving approximation and inversion of nonlinear functions, nonlinear functionals, and nonlinear operators, are cast in a reproducing kernel Hilbert space (RKHS) setting, and solved by orthogonal projection methods. The RKHS's used for the above purpose are the "arbitrarily weighted Fock spaces" introduced by de Figueiredo and Dwyer in II] and called in the present paper "generalized Fock (GF) spaces". These spaces consist of polynomials or power series in one or more scalar variables, or of finite (polynomic case) or infinite Volterra functional series in one or more functions, or of finite or infinite Volterra operator series in one or more functions. In each case, the space is equipped with an appropriate weighted inner product, with the option of making the choice of weights depend on the particular problem under consideration. These developments are illustrated by means of various applications, in particular, the modeling of semiconductor device characteristics, best approximation of nonlinear systems, and cancellation of large nonlinear distortion in signals propagating through electronic equipment.

A noise process with piecewise constant sample functions in the form of a finite random Walsh series is proposed, and its properties and potential usefulness in computations is analyzed. It is shown that the infinite series yields the familiar white noise model and that its integral yields a Wiener process almost surely. The integrated process also has all the properties of a Wiener process (modulo the sample points) in the finite series expansion. Using the stochastic calculus developed by McShane [10], [11], the necessary framework for handling stochastic system models driven by Walsh noise is developed.

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.

This paper reports that an analytic method is used to calculate the load responses of the two-wire transmission line excited by a plane-wave directly in the time domain. By the frequency-domain Baum–Liu–Tesche (BLT) equation, the time-domain analytic solutions are obtained and expressed in an infinite geometric series. Moreover, it is shown that there exist only finite nonzero terms in the infinite geometric series if the time variate is at a finite interval. In other word, the time-domain analytic solutions are expanded in a finite geometric series indeed if the time variate is at a finite interval. The computed results are subsequently compared with transient responses obtained by using the frequency-domain BLT equation via a fast Fourier transform, and the agreement is excellent.

The stress-focusing effect is the phenomenon that, under an instantaneous heating, stress waves reflected from the free surface of the cylinder result in very high stresses at the center. Ho solved the problem by using Laplace transform on time and presented the solution as infinite series summations, which converge very slowly for certain combinations of time and the radius of a cylinder. However, adopting a concept of the ray theory, the solution of stress waves in the cylinder is sorted out into rays according to the ray path of multiply reflected waves. The results expressed in an infinite series reveal that stresses peak out periodically at a constant period and the order of singularity of the stresses in a cylinder is O(ρ−2).

In this article, we discuss the concept of infinity (∞), infinitesimals □ , and a new mathematical operation that expands calculations into the realm of the infinite. This breakthrough allows for the computation of infinite numbers, leading to new possibilities for mathematical research and practical applications. By introducing infinite numbers, numerous advantages arise, such as accurately representing limits, divergences in integrals and infinite series, and extending calculations to infinity. This mathematical advancement opens doors to exploring uncharted territories, offering valuable insights and breakthroughs.

The evolution of indexing sequences and series has been little studied so far, although this topic presents difficulties for students. This symbolisation signified, however, an important means for operating more generally with finite and infinite series. This paper investigates the first steps during the eighteenth century and then the different processes in France and in Germany. In France, it was Lacroix with his textbooks from the end of the eighteenth and the early nineteenth century who paved the way, while dealing with difference series, for what would be established by Cauchy as a general practice with indexed sequences. In Germany, a pivotal role was played by the combinatorial school since the 1780s in establishing sophisticated indexing notations; the achievements by Dirksen in elaborating “higher-order” indexing notations prove to be decisive.

Addition theorems have been indispensable tools for the reduction of quantum transition amplitudes. They are normally utilized at the start of the process to move the angular dependence within plane waves, Coulomb potentials, and the like, into a sum over spherical harmonics that allows the angular integration to be carried out. These have historically been “two-range” addition theorems, characterized by the two-fold notation r>=Max[r1,r2] and r<=Min[r1,r2] and comprising a single infinite series. More recently, “one-range” addition theorems have been created that have no such piecewise notation, but at the cost of the introduction of another infinite series. We use a very different approach to derive an infinite set of addition theorems for Slater orbitals, hydrogenic and Hylleraas wave functions, and so on, that retain the one-range variable dependence but have, at worst, a finite second series rather than an infinite one. In addition, unlike previous addition theorems, they are applicable to more than one coordinate system. One of these addition theorems may also be used for Yukawa-like functions that may appear late in the reduction of amplitude integrals, and we show its utility for an integral that has stubbornly defied reduction to analytic form for nearly sixty years. Finally, we craft indefinite integrals of 15 half-integer Macdonald functions multiplied by (inverse) powers and negative exponentials containing squares of the integration variable.