Semi-convergent Series Research Articles

A modular-type formula for $$(x;q)_\infty $$ ( x ; q ) ∞

Let $$q=\text {e}^{2\pi i\tau }, \mathfrak {I}\tau >0$$ , $$x=\text {e}^{2\pi i{z}}$$ , $${z}\in \mathbb {C}$$ , and $$(x;q)_\infty =\prod _{n\ge 0}(1-xq^n)$$ . Let $$(q,x)\mapsto ({q_1},{x_1})$$ be the classical modular substitution given by the relations $${q_1}=\text {e}^{-2\pi i/\tau }$$ and $${x_1}=\text {e}^{2\pi i{z}/{\tau }}$$ . The main goal of this paper is to give a modular-type representation for the infinite product $$(x;q)_\infty $$ , this means, to compare the function defined by $$(x;q)_\infty $$ with that given by $$({x_1};{q_1})_\infty $$ . Inspired by the work (Stieltjes in Collected Papers, Springer, New York, 1993) of Stieltjes on semi-convergent series, we are led to a “closed” analytic formula for the ratio $$(x;q)_\infty /({x_1};{q_1})_\infty $$ by means of the dilogarithm combined with a Laplace type integral, which admits a divergent series as Taylor expansion at $$\log q=0$$ . Thus, the function $$(x;q)_\infty $$ is linked with its modular transform $$({x_1};{q_1})_\infty $$ in such an explicit manner that one can directly find the modular formulae known for Dedekind’s Eta function, Jacobi Theta function, and also for certain Lambert series. Moreover, one can remark that our results allow Ramanujan’s formula (Berndt in Ramanujan’s notebooks, Springer, New York, 1994, Entry 6’, p. 268) (see also Ramanujan in Notebook 2, Tata Institute of Fundamental Research, Bombay, 1957, p. 284) to be completed as a convergent expression for the infinite product $$(x;q)_\infty $$ .

Expansions of generalized Euler's constants into the series of polynomials in [formula omitted] and into the formal enveloping series with rational coefficients only

In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) γm are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in π−2 with rational coefficients and converges slightly better than Euler's series ∑n−2. The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's constants, which is more accurate than several well-known estimations. Finally, in Appendix A, the reader will also find two simple integral definitions for the Stirling numbers of the first kind, as well an upper bound for them.

Heat Conduction of Walls with a Monotone Temperature Change. Asymptotics and Quasi-Stationarity

Systematizing the partial solutions of the nonstationarity heat conduction problem of a flat wall in comparison with the general asymptotic solution of this problem, we have found the transverse temperature distributions with any monotone change in the ambient conditions and elucidated the heat conduction properties of the wall under these conditions. The asymptotic solution is given by semiconvergent series and definite integrals and has been investigated for power time dependences with an exponent of 0–2, which has enabled us to justify the concept of quasi-stationarity of the thermal parameters of the wall and obtain asymptotic errors and corrections defining the deviations of these parameters from their stationary values. The features of the average heat flows most resistant to thermal disturbances as to both time and amplitude have been considered.

On the modular behaviour of the infinite product [formula omitted

Let q = e 2 π i τ , ℑ τ > 0 , x = e 2 π i ξ ∈ C and ( x ; q ) ∞ = ∏ n ⩾ 0 ( 1 − x q n ) . Let ( q , x ) ↦ ( q ⁎ , ι q x ) be the classical modular substitution given by q ⁎ = e − 2 π i / τ and ι q x = e 2 π i ξ / τ . The main goal of this Note is to study the “modular behaviour” of the infinite product ( x ; q ) ∞ , this means, to compare the function defined by ( x ; q ) ∞ with that given by ( ι q x ; q ⁎ ) ∞ . Inspired by the work [16] of Stieltjes (1886) on some semi-convergent series, we are led to a “closed” analytic formula for the ratio ( x ; q ) ∞ / ( ι q x ; q ⁎ ) ∞ by means of the dilogarithm combined with a Laplace type integral, which admits a divergent series as Taylor expansion at log q = 0 . Thus, we can obtain an expression linking ( x ; q ) ∞ to its modular transform ( ι q x ; q ⁎ ) ∞ and which contains, in essence, the modular formulae known for Dedekindʼs eta function, Jacobi theta function and also for certain Lambert series. Among other applications, one can remark that our results allow one to interpret Ramanujanʼs formula (Berndt, 1994) [5, Entry 6, p. 265 & Entry 6′, p. 268] (see also Ramanujan, 1957 [10, pp. 365 & 284]) as being a convergent expression for the infinite product ( x ; q ) ∞ .

The discrete analogue of Laplace’s method

We give a justification of the discrete analogue of Laplace’s method applied to the asymptotic estimation of sums consisting of positive terms. The case considered is the series related to the hypergeometric function p F q − 1 ( x ) (with q ≥ p + 1 ) as x → + ∞ discussed by Stokes [G.G. Stokes, Note on the determination of arbitrary constants which appear as multipliers of semi-convergent series, Proc. Camb. Phil. Soc. 6 (1889) 362–366]. Two examples are given in which it is shown how higher order terms in the asymptotic expansion may be derived by this procedure.

Convergence properties of the intermolecular force series (1/R-expansion)

It is proven that the total energy of interacting molecular systems A and B at large intermolecular distancesR can be expanded in a semi-convergent series in powers of 1/R. It is further proven that “exchange forces” vanish faster than any power of 1/R.

A wave‐optical field solution for a line source over nonparallel stratified earth

We consider a wedge geometry for a model of a nonuniform layered earth. The plane wave spectrum of a line source excitation is allowed to reflect successively at the nonparallel interfaces. The semiconvergent series is truncated when the lowermost image term has been accounted for. The partial sum, so obtained, is considered to give a useful approximation to the resultant fields. It is found that there is a significant dependence on the tilt angle of the subsurface interface.

Die Berechnung der Nullstellen der Zylinderringfunktionen durch Potenzreihen

The zerosy=y n of the functionFκ(y)=J0(y) Y0(κy)−J0(κy) Y0(y) can be calculated with the semiconvergent series ofHankel, analoguous to the calculation of the zeros of the Bessel functions. The accuracy of this method is limited in itself and permits the calculation of only three places for higher values of κ [10].

A semi-convergent series with application to the collective theory of risk

Abstract 1. The Bessel function solution of the differential equation can be numerically calculated by means of the defining power series when x is small. For greater values of x it is more convenient to use an asymptotic expansion. In a paper in this journal, 1950, p. 16, I have deduced an integral expression from which an asymptotic expansion can be derived by developing (1 — t 2)−½ in series and integrating term by term (p. 18–19).

On the Product of Semi‐Convergent Series

Proceedings of the London Mathematical SocietyVolume s2-19, Issue 1 p. 57-74 Papers On the Product of Semi-Convergent Series T. S. Broderick, T. S. BroderickSearch for more papers by this author T. S. Broderick, T. S. BroderickSearch for more papers by this author First published: 1921 https://doi.org/10.1112/plms/s2-19.1.57AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volumes2-19, Issue11921Pages 57-74 RelatedInformation

135. [D. 2. a. β] On the proof of Riemann’s theorem on semi-convergent series

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On the discontinuity of arbitrary constants that appear as multipliers of semi-convergent series: A letter to the editor

On the discontinuity of arbitrary constants that appear as multipliers of semi-convergent series: A letter to the editor

Scientific Serial

American Journal of Mathematics, vol. xviii. No. 3. (Baltimore, July.)—On the multiplication and involution of semi-convergent series, by Prof. Cajori. In vol. xv. Prof. Cajori has generalised Voss's results (Math. Ann., vol. xxiv. p. 42), and some further contributions of his to this difficult subject are given in the Bulletin of the Am. Math. Soc. (vol. i. pp. 180–183). The search, he remarks, for expeditious tests on the applicability of Cauchy's multiplication rule to powers of semi-convergent series higher than the second power, has given rise to the present investigation, which begins with alternating semi-convergent series, and ends with certain trigonometric series.—Analytic functions suitable to represent substitutions, is an interesting following-up of a theorem due to Hermite (Comptes rendus, vol. lvii. p. 750), by L. E. Dickson. Further generalisations are promised in a dissertation by the author.—S. Kantor contributes an elaborate memoir, “Theorie der Transformationen im Rr, welche sich aus quadratischen zusammensetzen lassen,” which has as heading, “Boldness is caution in these circumstances.”—Tactical Memoranda, i.-iii., by E. H. Moore, is the opening one of a series of papers which the author proposes to publish, on certain more or less closely connected topics of tactic. He starts from Cayley's division of algebra into tactic and logistic. This instalment bears upon the work of Reze (Geometrie der Lage), S. Kantor, Klein, and many others; it also gives a generalisation of the fifteen-schoolgirls arrangement, and considers whist tournament arrangements, which are in ultimate formulation purely tactical.

On the Multiplication and Involution of Semi-Convergent Series

On the Multiplication and Involution of Semi-Convergent Series

Scientific Serials

Bulletin of the American Mathematical Society, vol. i. No. 7, (April 1895).—“Riemann and his significance for the development of modern mathematics,” is the translation, by A. Ziwet, of an address delivered by Prof. F. Klein at the general session of the Versammlung Deutscher Naturforscher und Aerzte in Vienna, September 27, 1894. In it the author attempts to give an idea of the life-work of Bernhard Riemann, “a man who more than any other has exerted a determining influence on the development of modern mathematics.”—Prof. Cajori contributes a note on the multiplication of semi-convergent series, in which, following up his work in a recent number of the Bulletin, he further extends results arrived at by Pringsheim (Math. Ann. vol. xxi. pp. 327—378) and by A. Voss (Math. Ann. vol. xxiv. pp. 42-47).—Mr. L. E. Dickson discusses Gergonne's Pile Problem (cf. Ball's “Recreations,” pp. 101-6), and points out one or two slight inaccuracies in a proof given by Dr. C. T. Hudson in Educational Times Reprints, vol. ix. pp. 89-91.—Prof. Ziwet gives an account of the Repertoire bibliographique des Sciences Mathematiques, i.e. a card catalogne of mathematical literature which has been widely circulated amongst mathematicians. Notes, and new publications, as usual, close the number.

The multiplication of semi-convergent series

The multiplication of semi-convergent series

On the Order of Terms in a Semi-Convergent Series

On the Order of Terms in a Semi-Convergent Series

On the Multiplication of Semi-Convergent Series

On the Multiplication of Semi-Convergent Series