The Moments of the Hydrogen Atom by the Method of Brackets

González, Iván; Kohl, Karen T.; Kondrashuk, Igor; Moll, Victor H.; Salinas, Daniel

doi:10.3842/sigma.2017.001

Cited by 7 publications

(9 citation statements)

References 29 publications

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“…Remark 1. 3 In the previous work [13] it was mentioned that it would be interesting to combine the Mellin-Barnes integrals and the method of brackets, due to the similarity of these methods. This combination is presented here.…”

Section: (): V-volmentioning

confidence: 99%

Mellin–Barnes integrals and the method of brackets

et al. 2022

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The method of brackets is a method for the evaluation of definite integrals based on a small number of rules. This is employed here for the evaluation of Mellin–Barnes integral. The fundamental idea is to transform these integral representations into a bracket series to obtain their values. The expansion of the gamma function in such a series constitute the main part of this new application. The power and flexibility of this procedure is illustrated with a variety of examples.

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Section: (): V-volmentioning

confidence: 99%

Mellin–Barnes integrals and the method of brackets

et al. 2022

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“…We focus here on yet another method which is the Method of Brackets (MoB) [2][3][4][5][6][7][8][9]. It is built on a set of heuristic rules, originally formulated to evaluate Feynman integrals [2,10,11], but which is also useful for the computation of certain definite integrals [3,4,12,13]. The MoB, which is based on a generalization of Ramanujan's Master Theorem [14,15], was proposed as an optimized version of the Negative Dimensional Integration Method (NDIM), the latter having been first introduced in [16,17] and applied to Feynman integrals in [18][19][20][21][22][23][24][25][26].…”

Section: Introductionmentioning

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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“…Remark 1.3. In the previous work [13] it was mentioned that it would be interesting to combine the Mellin-Barnes integrals and the method of brackets, due to the similarity of these methods. This combination is presented here.…”

Section: Introductionmentioning

confidence: 99%

Mellin-Barnes and the method of brackets

Gonzalez,

Kondrashuk,

Moll

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

The method of brackets is a method for the evaluation of definite integrals based on a small number of rules. This is employed here for the evaluation of Mellin-Barnes integral. The fundamental idea is to transform these integral representations into a bracket series to obtain their values. The expansion of the gamma function in such a series constitute the main part of this new application. The power and flexibility of this procedure is illustrated with a variety of examples.

show abstract

The Moments of the Hydrogen Atom by the Method of Brackets

Cited by 7 publications

References 29 publications

Mellin–Barnes integrals and the method of brackets

Mellin–Barnes integrals and the method of brackets

On the Method of Brackets

Mellin-Barnes and the method of brackets

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