Mellin–Barnes integrals and the method of brackets

González, Iván; Kondrashuk, Igor; Moll, Victor H.; Recabarren, Luis M.

doi:10.1140/epjc/s10052-021-09977-x

Cited by 6 publications

(5 citation statements)

References 24 publications

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“…We will now analyze the special case of the C n,k family with k = 1 using the Method of Brackets [2,3,20] and Mellin-Barnes representations. After this, each general integral with C n,k will be treated using the same procedure.…”

Section: Ising Integralsmentioning

confidence: 99%

“…Gradshyteyn and Ryzik [1] compiled a long list of such integrals. Recently there have been attempts to provide a derivation of a large number of these integrals, specifically the improper integral with limits from 0 to ∞ using the Original Method of Brackets (OMOB) [2][3][4][5][6][7]. Apart from this, some of the present authors have also evaluated the integral of quadratic and quartic types and their generalization using the OMOB, which has been reported in [8].…”

Section: Introductionmentioning

confidence: 99%

“…The new results for ∆ n (s) and J n with the above new results are also provided. Finally, we conclude the paper with some conclusions and possible future directions in section (6). In appendix C, we provide the table for all the MATHEMATICA files that we give and the packages required.…”

Section: Introductionmentioning

confidence: 99%

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Closed Form Expressions for Certain Improper Integrals of Mathematical Physics

B.¹,

Pathak²,

Sharma³

2023

Preprint

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We present new closed-form expressions for certain improper integrals of Mathematical Physics such as Ising, Box, and Associated integrals. The techniques we employ here include (a) the Method of Brackets and its modifications and suitable extensions and (b) the evaluation of the resulting Mellin-Barnes representations via the recently discovered Conic Hull method. Analytic continuations of these series solutions are then produced using the automated method of Olsson. Thus, combining all the recent advances allows for closed-form solutions for the hitherto unknown B 3 (s) and related integrals in terms of multivariable hypergeometric functions. Along the way, we also discuss certain complications while using the Original Method of Brackets for these evaluations and how to rectify them. The interesting cases of C 5,k is also studied. It is not yet fully resolved for the reasons we discuss in this paper.

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Section: Ising Integralsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Closed Form Expressions for Certain Improper Integrals of Mathematical Physics

B.¹,

Pathak²,

Sharma³

2023

Preprint

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“…This alternative approach was employed in [29] to show how some one-fold MB integrals can be computed from the MoB. Now, from Eq.…”

Section: Relation Between the Chmb And The Mobmentioning

confidence: 99%

“…We propose to perform this exercise in the context of a comparison of MoB against the Mellin-Barnes (MB) approach. It has been suggested several times in the past to study the links between these two methods and even to mix them [10,27,28] (see, by the way, the recent work [29] where MoB is used to compute some one-fold MB integrals). However, and this is the second aim of the present work, we will show that the significant progress that has been made in MB theory [30], by solving the important issue of deriving series representations of MB integrals of a very general form, shows the superiority of MB over MoB, since the former does not have any of the drawbacks, mentioned above, of the latter.…”

Section: Introductionmentioning

confidence: 99%

On the Method of Brackets

Ananthanarayan¹,

Banik²,

Friot³

et al. 2021

Preprint

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The Method of Brackets (MoB) is a technique used to compute definite integrals, that has its origin in the negative dimensional integration method. It was originally proposed for the evaluation of Feynman integrals for which, when applicable, it gives the results in terms of combinations of (multiple) series. We focus here on some of the limitations of MoB and address them by studying the Mellin-Barnes (MB) representation technique. There has been significant process recently in the study of the latter due to the development of a new computational approach based on conic hulls (see Phys. Rev. Lett. 127, 151601 (2021)). The comparison between the two methods helps to understand the limitations of the MoB, in particular when termwise divergent series appear. As a consequence, the MB technique is found to be superior over MoB for two major reasons: 1. the selection of the sets of series that form a series representation for a given integral follows, in the MB approach, from specific intersections of conic hulls, which, in contrast to MoB, does not need any convergence analysis of the involved series, and 2. MB can be used to evaluate resonant (i.e. logarithmic) cases where MoB fails due to the appearance of termwise divergent series. Furthermore, we show that the recently added Rule 5 of MoB naturally emerges as a consequence of the residue theorem in the context of MB.

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Hypergeometric structures in Feynman integrals

Blümlein

Saragnese

Schneider

2023

Ann Math Artif Intell

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For the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.

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Mellin–Barnes integrals and the method of brackets

Cited by 6 publications

References 24 publications

Closed Form Expressions for Certain Improper Integrals of Mathematical Physics

Closed Form Expressions for Certain Improper Integrals of Mathematical Physics

On the Method of Brackets

Hypergeometric structures in Feynman integrals

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