On the evaluation of the Appell F2 double hypergeometric function

Ananthanarayan, B.; Bera, Souvik; Friot, Samuel; Marichev, O. I.; Pathak, Tanay

doi:10.1016/j.cpc.2022.108589

Cited by 6 publications

(2 citation statements)

References 38 publications

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“…Simplified methods give analytical results that are more explicit and less restrictive [8]. The method of parentheses (MoB), proposed for evaluating Feynman integrals, gives results in terms of combinations of series(s), and it can also be used to prove the residue theorem [9]. Through A simple evaluation of a theta value and the Kronecker limit formula, we can verify complex functions not only by computing the integral by the residue theorem but also by using elliptic functions or analytical continuations [10].…”

Section: Discussionmentioning

confidence: 99%

Complex Analysis and Residue Theorem

Xia¹

2023

TNS

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The study of the properties of analytical functions is described as complex analysis. The residue theorem is an important conclusion in complex analysis. This paper introduces the origin of imaginary numbers from Cardano Formula defines the conversion of complex number formats from the Euler Theorem, and proves the intermediate theorem Cauchy Integral Formula before reaching our final conclusion and the goal of the paper, Residue Theorem. The name of the theorem comes from the concept of residue, which is defined using a functions Laurent series. We could then derive the Residue Theorem from the Cauchy Integral Theorem, also called the Cauchy-Goursat Theorem. We will be able to formalize our prior, ad hoc method of computing integrals over contours encompassing singularities. Additionally, it is a theorem that may be applied to zero-pole qualities and curved integral properties. The Residue Theorem is the basis of many essential mathematical facts revolving around line integrals, particularly in solving ODEs and PDEs and describing physics models.

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Section: Discussionmentioning

confidence: 99%

Complex Analysis and Residue Theorem

Xia¹

2023

TNS

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“…The series representation hence obtained, in general, can be written as hypergeometric functions or their derivatives. Independently, the issue of finding the analytic continuations (ACs) of the multivariable hypergeometric function using the method of Olsson [16,17], which has also been automated as a MATHEMAT-ICA package Olsson.wl [18] have been addressed recently. In this work, we show how these tools together, which were primarily directed at solving Feynman integrals, are of sufficient generality to find their use in the evaluation of the integrals considered here.…”

Section: Introductionmentioning

confidence: 99%

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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