Ramanujan’s Notebooks

Berndt, Bruce C.

doi:10.1007/978-1-4612-1088-7

Cited by 347 publications

(209 citation statements)

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“…Euler discovered this summation formula in connection with the so-called Basel problem, i.e., with determining ζ (2) in modern terminology. It can be found without proof in [23] (submitted 1732) and with a complete deduction in [24] (submitted 1735), where he used it to calculate ζ(2), ζ (3), ζ (4), and the Euler constant γ; for more details see [26]. The formula was found independently by Maclaurin in 1738 [38]; cf.…”

Section: Euler-maclaurin Summation Formula (Emsf)mentioning

confidence: 98%

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The Summation Formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their Interconnections with the Approximate Sampling Formula of Signal Analysis

et al. 2011

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This paper is concerned with the two summation formulae of Euler-Maclaurin (EMSF) and Abel-Plana (APSF) of numerical analysis, that of Poisson (PSF) of Fourier analysis, and the approximate sampling formula (ASF) of signal analysis. It is shown that these four fundamental propositions are all equivalent, in the sense that each is a corollary of any of the others. For this purpose ten of the twelve possible implications are established. Four of these, namely the implications of the grouping APSF ⇐ ASF ⇒ EMSF ⇔ PSF are shown here for the first time. The proofs of the others, which are already known and were established by three of the above authors, have been adapted to the present setting. In this unified exposition the use of powerful methods of proof has been avoided as far as possible, in order that the implications may stand in a clear light and not be overwhelmed by external factors. Finally, the four propositions of this paper are brought into connection with four After this manuscript was accepted for publication the authors learned that special issues were being prepared to honour Heinrich Wefelscheid on the occasion of his 70th anniversary. Since they valued his unstinted service to mathematics as a whole, thus the founding of the present journal and his leading editorship over the years, his publication of the Collected Works, in several volumes, of W. Blaschke and E. Landau, they expressed interest that the present paper, already dedicated to A. Ostrowski, be included in the special issues. 360 P. L. Butzer et al. Results. Math.propositions of mathematical analysis for bandlimited functions, including the Whittaker-Kotel'nikov-Shannon sampling theorem. In conclusion, all eight propositions are equivalent to another. Finally, the first three summation formulae are interpreted as quadrature formulae.Mathematics Subject Classification (2010). 65B15, 65D32, 94A20.

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Section: Euler-maclaurin Summation Formula (Emsf)mentioning

confidence: 98%

“…See also the review of Euler's life and works by Gautschi [27], and the overview by Apostol [3]. Of further interest is that Ramanujan [4,Chaps. 6,8] introduced a method of summation based on EMSF.…”

Section: Euler-maclaurin Summation Formula (Emsf)mentioning

confidence: 99%

The Summation Formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their Interconnections with the Approximate Sampling Formula of Signal Analysis

et al. 2011

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“…In the period 1900-1920, S. Ramanujan carried out extensive studies of the series (1.1), however, in most cases in unpublished notebooks without complete proofs (see [Ask1]). B. C. Berndt has published a series of edited notebooks [Be1], [Be2], [Be3] and [Be4]. Because complete reconstructed proofs are given there, Berndt has rescued this work of a genius from oblivion and obscurity and thus made many jewels of the mathematical science widely available.…”

Section: §1 Introductionmentioning

confidence: 99%

“…as r tends to 1 (see also [Ask1], [Be1], and [E]). Here and in the sequel, 6) and γ is the Euler-Mascheroni constant.…”

Section: §1 Introductionmentioning

confidence: 99%

Landen inequalities for hypergeometric functions

Qiu

Вуоринен²

1999

Nagoya Mathematical Journal

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“…To our knowledge, there apparently are few works in the literature that address the efficient computation of Clausen functions with n > 2. Recently, we were informed that a recurrence algorithm for the Clausen functions with n ≥ 2 can be constructed through Entry 13 in page 260 of [3]. In this paper, we obtain a different series, but equivalent to (1.1), for the Clausen functions Cl n (θ ) (n ≥ 2), which is not of the recurrence type and has an exponential convergence rate.…”

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confidence: 92%

An efficient calculation of the Clausen functions Cl n (θ)(n≥2)

Zhang

Liu

2009

Bit Numer Math

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The Clausen functions appear in many problems, such as in the computation of singular integrals, quantum field theory, and so on. In this paper, we consider the Clausen functions Cl n (θ ) with n ≥ 2. An efficient algorithm for evaluating them is suggested and the corresponding convergence analysis is established. Finally, some numerical examples are presented to show the efficiency of our algorithm.

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Ramanujan’s Notebooks

Cited by 347 publications

References 0 publications

The Summation Formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their Interconnections with the Approximate Sampling Formula of Signal Analysis

The Summation Formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their Interconnections with the Approximate Sampling Formula of Signal Analysis

Landen inequalities for hypergeometric functions

An efficient calculation of the Clausen functions Cl n (θ)(n≥2)

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