Asymptotic expansions of the Appell’s function 𝐹₁

Ferreira, Chelo; López, José L.

doi:10.1090/qam/2054598

Cited by 14 publications

(13 citation statements)

References 20 publications

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“…Their expansion can be shown to be a particular case of the expansion for F 1 given later in [11]. We will show below that both expansions (but not the error bounds!)…”

Section: Introductionmentioning

confidence: 81%

“…Finally, (39) is reduced to (11) with the help of the following formula found at http://functions.wolfram.com/07.27. 17.0029.01:…”

Section: Remark 3 Expansion [11 Formula (53)] Can Be Cast Into the mentioning

confidence: 99%

“…An asymptotic expansion for F 1 in the neighbourhood of infinity has been studied by Ferreira and López in [11]. Their expansion can be converted into an approximation around (1, 1) using the formula…”

Section: Introductionmentioning

confidence: 99%

“…can be obtained by simple rearrangement of (5) and use of known transformation formulas for F 1 . More precise approximations for F (λ, k) which cannot be reduced to expansions from [11] have been given recently by S.M. Sitnik and the author in [13].…”

Section: Introductionmentioning

confidence: 99%

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An approximation for zero-balanced Appell function F1 near (1,1)

Karp

2008

Journal of Mathematical Analysis and Applications

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We suggest an approximation for the zero-balanced Appell hypergeometric function F 1 near the singular point (1, 1). Our approximation can be viewed as a generalization of Ramanujan's approximation for zero-balanced 2 F 1 and is expressed in terms of 3 F 2 . We find an error bound and prove some basic properties of the suggested approximation which reproduce the similar properties of the Appell function. Our approximation reduces to the approximation of Carlson-Gustafson when the Appell function reduces to the first incomplete elliptic integral.

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“…Their expansion can be shown to be a particular case of the expansion for F 1 given later in [11]. We will show below that both expansions (but not the error bounds!)…”

Section: Introductionmentioning

confidence: 81%

“…Finally, (39) is reduced to (11) with the help of the following formula found at http://functions.wolfram.com/07.27. 17.0029.01:…”

Section: Remark 3 Expansion [11 Formula (53)] Can Be Cast Into the mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An approximation for zero-balanced Appell function F1 near (1,1)

Karp

2008

Journal of Mathematical Analysis and Applications

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“…However, the necessary asymptotic expansions of these functions are extremely complicated. We were able to find verifiably-correct expressions for the relevant asymptotics in some recent mathematics literature [70], but these expressions are very cumbersome to work with. It may be an interesting project in mathematical physics to figure out a way to obtain sufficiently transparent closed-form expressions for the asymptotics of these special functions for using them in QFT calculations, but it is beyond the scope of the present work.…”

Section: Jhep06(2014)046mentioning

confidence: 94%

Searching for Fermi surfaces in super-QED

Cherman

Grozdanov

Hardy

2014

J. High Energ. Phys.

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The exploration of strongly-interacting finite-density states of matter has been a major recent application of gauge-gravity duality. When the theories involved have a known Lagrangian description, they are typically deformations of large N supersymmetric gauge theories, which are unusual from a condensed-matter point of view. In order to better interpret the strong-coupling results from holography, an understanding of the weakcoupling behavior of such gauge theories would be useful for comparison. We take a first step in this direction by studying several simple supersymmetric and non-supersymmetric toy model gauge theories at zero temperature. Our supersymmetric examples are N = 1 super-QED and N = 2 super-QED, with finite densities of electron number and R-charge respectively. Despite the fact that fermionic fields couple to the chemical potentials we introduce, the structure of the interaction terms is such that in both of the supersymmetric cases the fermions do not develop a Fermi surface. One might suspect that all of the charge in such theories would be stored in the scalar condensates, but we show that this is not necessarily the case by giving an example of a theory without a Fermi surface where the fermions still manage to contribute to the charge density.

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Second-order transport, quasinormal modes and zero-viscosity limit in the Gauss-Bonnet holographic fluid

Grozdanov

Starinets

2017

J. High Energ. Phys.

129

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Gauss-Bonnet holographic fluid is a useful theoretical laboratory to study the effects of curvature-squared terms in the dual gravity action on transport coefficients, quasinormal spectra and the analytic structure of thermal correlators at strong coupling. To understand the behavior and possible pathologies of the Gauss-Bonnet fluid in 3 + 1 dimensions, we compute (analytically and non-perturbatively in the Gauss-Bonnet coupling) its second-order transport coefficients, the retarded two-and three-point correlation functions of the energy-momentum tensor in the hydrodynamic regime as well as the relevant quasinormal spectrum. The Haack-Yarom universal relation among the second-order transport coefficients is violated at second order in the Gauss-Bonnet coupling. In the zero-viscosity limit, the holographic fluid still produces entropy, while the momentum diffusion and the sound attenuation are suppressed at all orders in the hydrodynamic expansion. By adding higher-derivative electromagnetic field terms to the action, we also compute corrections to charge diffusion and identify the non-perturbative parameter regime in which the charge diffusion constant vanishes. 4 We use notations and conventions of [2]. See Appendix B and footnote 91 on page 128 of ref.[8] for clarification of sign conventions appearing in the literature. 5 Note that all transport coefficients in H are "dynamical" in terminology of ref. [19]. 6 As advertised in ref.[30] and shown below (and, independently, in ref. [48] using fluid-gravity duality methods), the Haack-Yarom relation does not hold non-pertutbatively in the Gauss-Bonnet coupling. 7 It appears that at weak coupling, the relation H = 0 does not hold. We briefly review the results at weak coupling in Appendix A. 8 We would like to thank P. Kovtun and M. Rangamani for a discussion of these issues.

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Asymptotic expansions of the Appell’s function 𝐹₁

Abstract: The first Appell’s hypergeometric function F 1 ( a , b , c , d ; x , y ) {F_1}\left ( a, b, c, d; x, y \right ) is considered for large values of its variables x x and/or y y . An integral representation of F 1 … Show more

Cited by 14 publications

References 20 publications

An approximation for zero-balanced Appell function F1 near (1,1)

An approximation for zero-balanced Appell function F1 near (1,1)

Searching for Fermi surfaces in super-QED

Second-order transport, quasinormal modes and zero-viscosity limit in the Gauss-Bonnet holographic fluid

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