Moments of products of elliptic integrals

Wan, J. G.

doi:10.1016/j.aam.2011.04.007

Cited by 22 publications

(49 citation statements)

References 13 publications

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“…The results given in the present subsection are inspired in part by those in [19] on the moments of products of complete elliptic integrals. We remark that 70 the contiguity relations for hypergeometric functions can be applied to obtain similar results.…”

Section: On the Moments Of Elliptic-type Integralsmentioning

confidence: 99%

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On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier–Legendre expansions

Campbell

D’Aurizio

Sondow

2019

Journal of Mathematical Analysis and Applications

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Motivated by our previous work on hypergeometric functions and the parbelos constant, we perform a deeper investigation on the interplay among generalized complete elliptic integrals, Fourier-Legendre (FL) series expansions, and p F q series. We produce new hypergeometric transformations and closed-form evaluations for new series involving harmonic numbers, through the use of the integration method outlined as follows: Letting K denote the complete elliptic integral of the first kind, for a suitable function g we evaluate integrals such as 1 0 * Corresponding author

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Section: On the Moments Of Elliptic-type Integralsmentioning

confidence: 99%

“…Using the Maclaurin series for K in the integrand in (19), we see that the definite integral in Lemma 5.2 must be equal to π 2 i∈N0 1 16…”

Section: On the Moments Of Elliptic-type Integralsmentioning

confidence: 99%

On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier–Legendre expansions

Campbell

D’Aurizio

Sondow

2019

Journal of Mathematical Analysis and Applications

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“…is assumed. This integral, which can also be written in terms of kc or k ′ c , has been found nowhere in the literature (however, see Prudnikov et al 1988;Wan 2012). It must therefore be estimated numerically.…”

Section: Shells With a Thick Edge Ie Tubesmentioning

confidence: 99%

Interior potential of a toroidal shell from pole values

Huré

Trova

Karas

et al. 2019

Monthly Notices of the Royal Astronomical Society

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We have investigated the toroidal analog of ellipsoidal shells of matter, which are of great significance in Astrophysics. The exact formula for the gravitational potential Ψ(R, Z) of a shell with a circular section at the pole of toroidal coordinates is first established. It depends on the mass of the shell, its main radius and axis-ratio e (i.e. core-to-main radius ratio), and involves the product of the complete elliptic integrals of the first and second kinds. Next, we show that successive partial derivatives ∂ n+m Ψ/∂ R n ∂ Z m are also accessible by analytical means at that singular point, thereby enabling the expansion of the interior potential as a bivariate series. Then, we have generated approximations at orders 0, 1, 2 and 3, corresponding to increasing accuracy. Numerical experiments confirm the great reliability of the approach, in particular for small-to-moderate axis ratios (e 2 0.1 typically). In contrast with the ellipsoidal case (Newton's theorem), the potential is not uniform inside the shell cavity as a consequence of the curvature. We explain how to construct the interior potential of toroidal shells with a thick edge (i.e. tubes), and how a core stratification can be accounted for. This is a new step towards the full description of the gravitating potential and forces of tori and rings. Applications also concern electrically-charged systems, and thus go beyond the context of gravitation.

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“…James Wan [83] has been able to prove many of these identities, including (6), but by no means all. A followup work [84] contains remarkable identities, such as…”

Section: Moments Of Elliptic Integral Functionsmentioning

confidence: 99%

High-Precision Arithmetic in Mathematical Physics

Bailey

Borwein

2015

Mathematics

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For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This article discusses the challenge of high-precision computation, in the context of mathematical physics, and highlights what facilities are required to support future computation, in light of emerging developments in computer architecture.

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Moments of products of elliptic integrals

Abstract: We consider the moments of products of complete elliptic integrals of the first and second kinds. In particular, we derive new results using elementary means, aided by computer experimentation and a theorem of W. Zudilin. Diverse related evaluations, and two conjectures, are also given.

Cited by 22 publications

References 13 publications

On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier–Legendre expansions

On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier–Legendre expansions

Interior potential of a toroidal shell from pole values

High-Precision Arithmetic in Mathematical Physics

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