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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf,len=784
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='13436v1  [math-ph]  31 Jan 2023  Closed Form Expressions for Certain Improper Integrals of  Mathematical Physics  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Ananthanarayan * Tanay Pathak† Kartik Sharma‡  Centre for High Energy Physics, Indian Institute of Science,  Bangalore-560012, Karnataka, India  Abstract  We present new closed-form expressions for certain improper integrals of Mathematical Physics such as Ising, Box,  and Associated integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Analytic continuations of these series solutions are then produced using the automated method  of Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Thus, combining all the recent advances allows for closed-form solutions for the hitherto unknown B3(s)  and related integrals in terms of multivariable hypergeometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Along the way, we also discuss certain com-  plications while using the Original Method of Brackets for these evaluations and how to rectify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The interesting  cases of C5,k is also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' It is not yet fully resolved for the reasons we discuss in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  1  Introduction  In studies of theoretical physics and mathematics, various integrals appear whose symbolic evaluation is sought  after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Gradshyteyn and Ryzik [1] compiled a long list of such integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 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) [2–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  In the present investigation, we turn to other interesting improper integrals that appear in Mathematical  Physics, such as the Ising integrals and the Box integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Our work is motivated by the need to express them in  terms of elegant closed-form expression or in terms of known functions of mathematical physics, especially the hyper-  geometric functions [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' In the recent past, several tools have also been developed to facilitate tasks of symbolic  evaluation of these integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Our results here have been facilitated by the recent development of tools and ad-  vances in various theoretical treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Note for instance, the recently proposed solution to the problem of finding  the series solution of the N-dimensional Mellin-Barnes (MB) representation [11–13], using what has been termed  as the Conic Hull Mellin Barnes (CHMB) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' This has also been automated as the MATHEMATICA package  MBConichulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The series representation hence obtained, in general, can be written as hypergeometric  functions or their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Independently, the issue of finding the analytic continuations (ACs) of the multivari-  able hypergeometric function using the method of Olsson [16,17], which has also been automated as a MATHEMAT-  ICA package Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl [18] have been addressed recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  We will consider the Ising integrals which have been studied in the Ising model [19–22] and also have been in  the context of OMOB [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Apart from the evaluation with these newly developed tools, we will also consider certain  complications while doing similar evaluations with the OMOB [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' One of them is the use of regulators for the  evaluation of the Ising integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' This arises in the case of Ising integrals C3,1 and C4,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' For the case of C4,1, it is  further complicated due to the use of two regulators, which, when the proper limiting procedure is applied, will give  the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' However, we point out that such a procedure is complicated and thus use the Modified Method of  Brackets (MMOB) [24] to get the MB-integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' This MB integral can then be evaluated without any introduction  of such regulators and thus provides an efficient way to deal with these integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Using a similar procedure, we  anant@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='in  †tanaypathak@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='in  ‡kartiksharma@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='in  1  attempt to evaluate the elusive C5,k integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' However, we hit a roadblock for the same, as the resulting series does  not converge and would require a proper analytic continuation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' At present, we find this task beyond the  reach of the tools at hand, though we provide a possible way to achieve the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Yet such results still shed some light  on the form that these integrals can be evaluated to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' All the results are provided in the ancillary MATHEMATICA file  Ising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='nb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Box integrals [25–28] are another interesting integrals where such techniques can be applied to get new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  They do carry a physical meaning in the sense that they provide the expected distance between two randomly chosen  points over the unit n-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We consider the two special cases of them, namely the Bn(s) and the ∆n(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We use  the same techniques and derive the closed form results for already known B1(s) and B2(s) and new evaluation for  B3(s) and B4(s) for general values of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The results are in terms of multi-variable hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' These  evaluations further require the use of an analytic continuation procedure which has been done using Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  All the results are provided in the ancillary MATHEMATICA file Box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='nb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' These results for box integrals can then  be further used to evaluate the Jellium potential Jn, which can be related to box integral Bn(s) [26, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Finally,  we give a general MB integral for Bn(s), which can be used to find the closed form result for all values of n and  s using MbConicHull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' With all this, we find new connections between the Box integrals and the multivariable  hypergeometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' All our calculations rely heavily on MATHEMATICA as we try to achieve the symbolic  results for all the problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  The paper is structured as follows: In section (2) using an example given in [4], we point out the problem in the  OMOB and discuss the alternative to surpass this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We then, in section (3), proceed to the evaluation of Ising  integrals up to n = 4 while contrasting our method with the method used before to achieve the same in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' In section  (4) we attempt to solve the C5,k integral and point out a general integral C5,k(α,β) which gives C5,k as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Though we point out that it is not the final result, a proper analytic continuation procedure is required to get C5,k  from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We then evaluate box integral Bn(s) for n = 3,4 in section (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The new results for ∆n(s) and Jn with the above  new results are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Finally, we conclude the paper with some conclusions and possible future directions  in section (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' In appendix C, we provide the table for all the MATHEMATICA files that we give and the packages  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  2  Method of Brackets revisited  We will first illustrate the OMOB using a simple example of integral evaluation as given in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We will first evaluate  the integral by directly using the OMOB, then briefly propose a possible resolution while doing such evaluations, and  then illustrate the alternative method to do the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  We consider the following integral  H1(a,b) =  �∞  0  K0(ax)K0(bx)  (1)  The integral is introduced to facilitate the evaluation of another integral, which is given by putting a = b  H(a) =  �∞  0  K2  0(ax)dx  (2)  We can express K0(x) using the following series expansion:  K0(ax) =  �  n1  φn1  a2n1Γ(−n1)  22n1+1  x2n1  (3)  where φn = (−1)n  Γ(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  This expansion uses a divergent series, and we can express the result in the form of an integral representation  as  K0(bx) = 1  2  �∞  0  exp  �  −t− b2x2  4t  � dt  t  (4)  Using the OMOB, we get:  K0(bx) =  �  n2,n3  φn2,n3  b2n3 x2n3  22n3+1 〈n2 − n3〉  (5)  Substituting the bracket series in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (1), we get  H1(a,b) =  �  n1,n2,n3  φn1,n2,n3  a2n1b2n3Γ(−n1)  22n1+2n3+2  〈n2 − n3〉〈2n1 +2n3 +1〉  (6)  2  Now, we need to solve the bracket equations, which involve 2 equations but 3 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Evaluating this we get  following 3 series, Ti where ni is the free variable:  T1 = 1  4a  �  n  φnΓ(−n)Γ2  �  n+ 1  2  �� b  a  �2n  T2 = 1  4a  �  n  φnΓ(−n)Γ2  �  n+ 1  2  �� b  a  �2n  T3 = 1  4a  �  n  φnΓ(−n)Γ2  �  n+ 1  2  �� b  a  �2n  (7)  Using the rules of the OMOB, all the 3 series of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (7) have to be discarded as they are divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  A solution to such a problem, as implemented in [4], is to regularize the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' This amounts to modifying  the bracket 〈n2−n3〉 → 〈n2−n3+ǫ〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' With this modification, when n1 is a free variable, one gets the series that contains  Γ(−n), which is diverging and is thus discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' While for the other cases, one gets two series with ǫ parameter (in  the form of Γ(−n + ǫ) and Γ(−n − ǫ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' In these series, when the proper limiting procedure is done, along with the  condition a = b to ease the calculation, they give the result for the integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Thus, the original integral of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (1) we started with still remains elusive, as the calculation is much more involved (the limiting procedure) within  this present framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  An alternative to the above evaluation, free from choosing the regulator and doing the tedious limiting procedure,  is to use the MB representation derived using the MMOB [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Using it, we get the following MB representation for  the integral given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (1)  H1(a,b) = 1  4  c+i∞  �  c−i∞  dz  2πi a−2z−1b2zΓ(−z)2Γ  �1  2(2z +1)  �2  (8)  The above MB integral can be readily evaluated in MATHEMATICA to give the following result  H1(a,b) =  π  �  a2  b2 K  �  1− a2  b2  �  2a  (9)  where K(x) is the complete elliptic integral of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Thus we get the value of the original integrals, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (1) we  started with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  For the special case of a = b, using K(0) = π  2 we get  H1(a,a) = H(a) = π2  4a  (10)  So we see that for the simple cases, too, using the MB representation to evaluate these integrals provides an efficient  way to evaluate these integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  3  Ising integrals  In this section, we will analyze the integrals of the “Ising class".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Ising models are extensively used to study the  statistical nature of ferromagnets [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The model accounts for the magnetic dipole moments of the spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The n -  dimensional integrals are denoted by Cn,Dn,En, where Dn is found in the magnetic susceptibility integrals essential  to the Ising calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Dn = 4  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  �∞  0  ···  �∞  0  �  i<j  � ui−u j  ui+u j  �2  (�n  j=1(u j +1/u j))2  du1  u1  ··· dun  un  (11)  The integral Dn provides great insights into the symmetry breaking at low-temperature phase and finds great use in  Quantum Field Theories and condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' However, it is difficult to evaluate these integrals computa-  tionally and analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' On the other hand, the Cn (Cn = Cn,1) class integrals which are closely related to the Dn  class, are easier to tackle and can produce closed-form expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  The general Ising integrals Cn,k is defined as  Cn,k = 4  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  �∞  0  ···  �∞  0  1  (�n  j=1(u j +1/u j))k+1  du1  u1  ··· dun  un  (12)  3  The above expression can also be expressed as the moments of power of Bessel Function K0 as  Cn,k = 2n−k+1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' cn,k := 2n−k+1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  �∞  0  tkKn  0 (t)dt  (13)  We will now analyze the special case of the Cn,k family with k = 1 using the Method of Brackets [2,3,20] and Mellin-  Barnes representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' After this, each general integral with Cn,k will be treated using the same procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The  C1,k and C2,k integrals are easily tractable, and the results for them have been given just for completeness’ sake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The  problem occurs when one considers Cn,k for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Below we use the MMOB [24] and show that for the evaluation of  the integrals requiring the use of regulators, it is better to use the MMOB and solve the corresponding integral using  the CHMB method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The main utility of the method is that the limiting procedure is automatically taken care of while  finding the residue in the case of CHMB, which is at times difficult, especially when there is more than 1 regulator,  as in the case of C4,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1  C1,k  For n = 1, we have  C1,k = 4  1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  �∞  0  1  (u1 +1/u1)k+1  du1  u1  (14)  The integral can simply be evaluated to give the general closed form:  C1,k =  �π21−kΓ  �  k+1  2  �  Γ  �  k  2 +1  �  (15)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2  C2,k  For k = 1, we get:  C2,k = 4  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  �∞  0  �∞  0  1  (u1 +1/u1 + u2 +1/u2)k+1  du1  u1  du2  u2  (16)  This evaluation using the MOB, for k = 1, gives:  C2,1 = 1  (17)  The integral for the general value of k can also be evaluated to give the following closed form:  C2,k =  Γ  �  k  2 + 1  2  �4  Γ(k+1)2  (18)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='3  C3,k and C3,k(α,β,γ)  For k = 1, we get:  C3,1 = 4  3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  �∞  0  �∞  0  �∞  0  1  (u1 +1/u1 + u2 +1/u2 + u3 +1/u3)2  du1  u1  du2  u2  du3  u3  (19)  We will illustrate the problem encountered in OMOB by writing the bracket series for the generalized case C3,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Taking k = 1 will give us the result for C3,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  The following form of the integrand is motivated to maximize the number of brackets series in the expansion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' which  in turn reduces the number of variables:  C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='k = 2  3  �∞  0  �∞  0  �∞  0  (u1u2u3)k  (u1u2u3(u1 + u2)+ u3(u1 + u2)+ u1u2u2  3 + u1u2)k+1 du1du2du3  (20)  Expanding the denominator using the rules of MOB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  �  {n}  φ{n}(u1u2)n1+n3+n4 zn1+n2+2n3(u1 + u2)n1+n2 〈k+1+ n1 + n2 + n3 + n4〉  Γ(k+1)  (21)  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (u1 + u2)n1+n2 has to be further expanded as:  (u1 + u2)n1+n2 =  �  n5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='n6  φn5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='n6un5  1 un6  2  〈−n1 − n2 + n5 + n6〉  Γ(−n1 − n2)  (22)  4  Combining the expansions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' the C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='k integral takes the form:  C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='k =  2  3Γ(k+1)  �  {n}  φ{n}  〈−n1 − n2 + n5 + n6〉  Γ(−n1 − n2)  (23)  ×〈k+1+ n1 + n3 + n4 + n5〉〈k+1+ n1 + n3 + n4 + n6〉  ×〈k+1+ n1 + n2 +2n3〉〈k+1+ n1 + n2 + n3 + n4〉  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' the rules of MOB demand that we solve the linear equations of the brackets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' but that poses the problem of  giving rise to divergent terms like Γ(−n) and renders the whole procedure useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' To solve the issue, it is suggested  to introduce regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' For the case of C3,k, one regulator is enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' In particular, ǫ(→ 0) is introduced in the bracket  as 〈k+1+n1+n2+2n3〉 → 〈k+1+n1+n2+2n3+ǫ〉 which mimics the effect of introducing a factor of uǫ  3 in the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Now, with this “new" bracket series, the divergent terms take the form of Γ(−n −ǫ) and are easier to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' In  the regime of OMOB, one requires the expansion of Γ(x) around integers to deal with the problem, which increases  the complexity of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  As n increases, the number of regulators increases monotonically and complicates the limiting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' On  the other hand, MMOB doesn’t call for any regulators and is very computationally friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Using the MMOB in the  above bracket series, we get the following MB representation for the C3,1  C3,1 = 1  3  c+i∞  �  c−i∞  dz  2πi  Γ(−z)4 Γ(1+ z)2  Γ(−2z)  (24)  This evaluates to  C3,1 = 2  27  �  6i  �  3  �  Li2  �  1  4 − i  �  3  4  �  −Li2  �  i  �  3  4  + 1  4  ��  +π  �  3log(4)−ψ(1)  �1  3  �  +ψ(1)  �2  3  ��  (25)  where ψ(1) is the polygamma function of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  The generalized integral C3,k can be similarly obtained using the MMOB to give the following MB representation:  C3,k =  1  3Γ(k+1)  c+i∞  �  c−i∞  dz  2πi  Γ(−z)4 Γ  � 1  2(k+2z +1)  �2  Γ(−2z)  (26)  The above integral can be evaluated to give  C3,k =  2  3k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  �  πG2,3  3,3  �1  4  ����  1,1,1  k+1  2 , k+1  2 , 1  2  �  (27)  where G is the Meijer-G function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  A further generalization of C3,k integral namely C3,k(α,β,γ) is given in [3] where the following integral is considered  C3,k(α,β,γ) =  �∞  0  �∞  0  �∞  0  xα−1yβ−1zγ−1  (x+1/x+ y+1/y+ z +1/z)k+1 dxdydz  (28)  Using the MMOB, we get the following MB representation  C3,k(α,β,γ) =  1  3Γ(k+1)  c+i∞  �  c−i∞  dz  2πi  Γ(−z)Γ(−z +α−1)Γ(−z −β+1)Γ(−z +α−β)Γ  �1  2(k+2z −α+β−γ+2)  �  Γ  � 1  2(k+2z −α+β+γ)  �  Γ(−2z +α−β)  (29)  The result is given in the MATHEMATICA file Ising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='nb and is found to be :  = − 1  3k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='π3/2 csc(πγ)2−γ−k−1�  4γΓ  �1  2(k−α−β−γ+4)  �  Γ  �1  2(k+α−β−γ+2)  �  Γ  �1  2(k−α+β−γ+2)  �  Γ  �1  2(k+α+β−γ)  �  (30)  × 4 ˜F3  �1  2(k+α+β−γ), 1  2(k−α−β−γ+4), 1  2(k+α−β−γ+2), 1  2(k−α+β−γ+2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1  2(k−γ+2), 1  2(k−γ+3),2−γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1  4  �  −4Γ  �1  2(k−α−��+γ+2)  �  Γ  �1  2(k+α−β+γ)  �  Γ  �1  2(k−α+β+γ)  �  Γ  �1  2(k+α+β+γ−2)  �  × 4 ˜F3  �1  2(k−α+β+γ), 1  2(k+α+β+γ−2), 1  2(k−α−β+γ+2), 1  2(k+α−β+γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' k+γ  2  , 1  2(k+γ+1),γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1  4  ��  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='4  C4,k and C4,k(α,β,γ,δ)  For k = 1:  C4,1 = 4  4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  �∞  0  �∞  0  �∞  0  �∞  0  1  (u1 +1/u1 + u2 +1/u2 + u3 +1/u3 + u4 +1/u4)2  du1  u1  du2  u2  du3  u3  du4  u4  (31)  If one proceeds with the OMOB as in the case of C3,1, one is now required to use 2 regulators, namely ǫ and A [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The  result for C4,1 is then obtained by taking the limit ǫ → 0, A→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The use of two regulators significantly complicates  the task of doing the limiting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' So we again proceed with the use of the MMOB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Using the MOB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' we get the  following MB representation for C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1:  C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1 = 1  12  c+i∞  �  c−i∞  dz  2πi  Γ(−z)4 Γ(1+ z)4  Γ(−2z) Γ(2+2z)  (32)  This can be evaluated to give  C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1 = 7ζ(3)  12  (33)  The general case for n = 4 can be simplified to the following MB representation:  C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='k =  1  12Γ(k+1)  c+i∞  �  c−i∞  dz  2πi  Γ(−z)4 Γ  �  k+1  2 + z  �4  Γ(−2z) Γ(k+2z +1)  (34)  This can be evaluated to give the closed-form expression:  C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='k = π 2−k−1  3Γ(k+1)G3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='3  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='4  �  1  ����  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' k+2  2  k+1  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' k+1  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' k+1  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1  2  �  (35)  The given expression is of particular interest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' as seen from its values when evaluated for any odd values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' When  C4,k is evaluated for any odd k, it takes the form of aζ(3) + b function, where a and b are some rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Some of the values are provided for reference in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  A further generalization of C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='k integral namely C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='k(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='δ) can be considered as follows  C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='k(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='δ) =  �∞  0  �∞  0  �∞  0  �∞  0  xα−1yβ−1zγ−1wδ−1  (x+1/x+ y+1/y+ z +1/z + w+1/w)k+1 dxdydzdw  (36)  Using the MMOB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' we get the following MB representation  C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='k(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='γ) =  1  12Γ(k+1)  c+i∞  �  c−i∞  dz  2πi  Γ(−z)Γ(−z +γ−1)Γ(−z −δ+1)Γ(−z +γ−δ)Γ  �1  2(k+2z −α−β−γ+δ+3)  �  12Γ(k+1)Γ(−2z +γ−δ)Γ  � 1  2(k+2z −α−β−γ+δ+3)+ 1  2(k+2z +α+β−γ+δ−1)  �  ×Γ  �1  2(k+2z +α−β−γ+δ+1)  �  Γ  �1  2(k+2z −α+β−γ+δ+1)  �  Γ  �1  2(k+2z +α+β−γ+δ−1)  �  (37)  The above integral can be evaluated as before,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' and the solution has been provided in the accompanying MATHEMAT-  ICA file Ising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  We end this section by noting that given an integral, the evaluation of its MB representation obtained using  the MMOB [24] is more efficient than using the OMOB and its rules to evaluate the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The regulators and the  limiting procedure in the OMOB are automatically taken care of in the evaluation of MB integrals while evaluating  the residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Alternatively, this suggests that one can try to find a better rule that concerns the elimination of the  bracket for the OMOB so that one does not require regulators and the result is obtained with their use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  4  An attempt at C5,k  Using the machinery developed so far, we now attempt to evaluate the C5 integral in the same spirit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Using the MOB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  we get the following MB representation for C5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='k  C5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='k =  1  60Γ(k+1)  c1+i∞  �  c1−i∞  dz1  2πi  c2+i∞  �  c2−i∞  dz2  2πi  Γ(−z1)4Γ(−z2)4Γ  � 1  2 (k+2z1 +2z2 +1)  �2  Γ(−2z1)Γ(−2z2)  (38)  6  k  C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='k  0  1  6πG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='3  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='4  �  1  ����  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1  1  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1  2  �  1  7ζ(3)  12  2  1  48πG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='3  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='4  �  1  ����  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2  3  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 3  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 3  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1  2  �  3  7ζ(3)−6  1152  4  1  2304πG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='3  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='4  �  1  ����  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='3  5  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 5  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 5  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1  2  �  5  49ζ(3)−54  368640  6  1  276480πG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='3  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='4  �  1  ����  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='4  7  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 7  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 7  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1  2  �  7  63ζ(3)−74  15482880  Table 1: Values of C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='k for k = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='7  Evaluation of the above integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' when done directly using the MBConicHulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl, would result in the divergent  series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' A suitable way to approach such evaluation would be by taking two parameters that serve as the variables  for the series that appear and then evaluating the results with these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' For the C5,k integral we have the  following evaluation  C5,k(α,β) =  1  60Γ(k+1)  c1+i∞  �  c1−i∞  dz1  2πi  c2+i∞  �  c2−i∞  dz2  2πi (α)z1(β)z2 Γ(−z1)4Γ(−z2)4Γ  � 1  2 (k+2z1 +2z2 +1)  �2  Γ(−2z1)Γ(−2z2)  (39)  We notice that the integral Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (39) has a more general structure than the integral Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (38) with the introduction of  the two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The C5,k can be obtained by putting α = β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The evaluation of the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (39) has been done in  the accompanying MATHEMATICA file Ising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  We also note that though we have a result for integral (39), the result is not convergent for the value of interest  α = β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Proper analytic continuation techniques have to be used to achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' At present, with the form of  series that we obtain, the task is not achievable using Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' With the form of series at hand we believe that it  can be written as a derivative of ‘some’ hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Then Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl can be used to find the ACs of this  hypergeometric function so that it converges for α = β = 1, and then the derivative can be performed to get the final  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  7  5  Box Integrals  For dimension n, we define the box integral as the expected distance from a fixed point q (can be origin also) of point  r chosen randomly and independently over the unit n-cube, with parameter s,  Bn(s) =  �1  0  ···  �1  0  �  (r1)2 +···+(rn)2�s/2  dr1 ···drn  (40)  ∆n(s) =  �1  0  ···  �1  0  �  (r1 − q1)2 +···+(rn − qn)2�s/2  dr1 ···drndq1 ···dqn  (41)  For certain special values of parameter s, the above integrals give the following interpretation:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Bn(1): It gives the expected distance from the origin for a random point of the n-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' ∆n(1): It gives the expected distance between two random points of the n-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Due to the physical significance of the box integrals and also their use in the electrostatic potential calculations, we  wanted to evaluate these integrals and give closed-form expressions using the Method of Brackets that has been im-  plemented throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Using the quadrature formulae for all complex powers [25,26,29,33,34], we use the functions:  b(u) =  �1  0  e−u2x2dx =  �πerf(u)  2u  (42)  d(u) =  �1  0  �1  0  e−u2(x−y)2dydx =  �πuerf(u)+ e−u2 −1  u2  (43)  which gives us the relation:  Bn(s) =  2  Γ(−s/2)  �∞  0  u−s−1bn(u)du  (44)  ∆n(s) =  2  Γ(−s/2)  �∞  0  u−s−1dn(u)du  (45)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1  Bn(s)  Now, for the method of brackets to be operational, we need integrals of the form with limits from 0 to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We need to  make an Euler substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The following substitution has been found to be the most efficient:  x →  a  1+ a  (46)  which makes the integral  b(u) =  �1  0  e−u2x2dx =  �∞  0  e−u2�  a  1+a  �2  1  (1+ a)2 da  (47)  b(u) =  �∞  0  ∞  �  n=0  1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  � −u2a2  (1+ a)2  �n  1  (1+ a)2 da  (48)  Substituting this back in Bn(u) and applying MMOB, it is obtained that Bn(s) has a pole at s = −n and we finally  get:  B1(s) =  1  s+1,s ̸= −1  (49)  B2(s) =  2  s+2 2F1  �1  2,− s  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 3  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='−1  �  ,s ̸= −2  (50)  The first two cases were easy to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The first non-trivial evaluation is that of B3(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We found two different  results for the same by using two different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Firstly we consider the following representation of B3 [26]:  B3(s) =  3  3+ s C2,0(s,1) =  6  (3+ s)(2+ s)  �π/4  0  ��  1+sec2 t  �s/2+1 −1  �  (51)  8  The above can interestingly be evaluated in MATHEMATICA using Integrate command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Using it, we get the  following evaluation for the B3(s) integral  B3(s) =  6  (s+2)(s+3)  �  iF1  �  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1  2,− s  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2,−2  �  − 2  s+1  2  s+1 F1  �1  2(−s−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='−1  2,− s  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1− s  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1  2,−1  2  �  − i 2F1  �  1,− s  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 3  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='−1  �  +2s/2 2F1  �1  2,− s  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 3  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='−1  2  �  −  �π  4Γ  �  1− s  2  � 2F1  �  − s  2 − 1  2,− s  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='1− s  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='−1  �  Γ  �  − s  2 − 1  2  �  − π  4  �  (52)  where F1(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='b1,b2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='x, y) is the Appell F1 function which is defined for |x| < 1∧|y| < 1 as:  F1(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='b1,b2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='x, y) =  ∞  �  m,n=0  (a)m+n(b1)m(b2)n  (c)m+nm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  xmyn  (53)  where (q)n is the Pochhammer symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (52) requires the evaluation of the Appell F1 outside its region of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Such evaluation requires  the use of analytic continuation of F1, which has been done by Olsson [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Though we got the result using MATHEMATICA , it doesn’t provide many insights so as to aid the computations of  other Bn(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' So we proceed to a more systematic evaluation of the B3(s) so that the results can be generalized to other  values of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Using the MMOB [24] we get the following Mellin-Barnes integral for the B3(s)  B3(s) =  c1+i∞  �  c1−i∞  c2+i∞  �  c2−i∞  Γ(−z1)Γ(−z2)Γ(2z1 +1)Γ(2z2 +1)Γ(s−2z1 −2z2 +1)Γ  �  − s  2 + z1 + z2  �  Γ  �  − s  2  �  Γ(2z1 +2)Γ(2z2 +2)Γ(s−2z1 −2z2 +2)  dz2  2πi  dz1  2πi  (54)  We evaluate the above integral using the MBConicHulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl package [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The evaluation gives the following result:  B3(s) = −  π  2  �  s2 +5s+6  � +  �π  �  (s+2)2F1  � 1  2,− s  2 − 1  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 3  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='−1  �  + 2F1  �  − s  2 −1,− s  2 − 1  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='− s  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='−1  ��  Γ  �  − s  2 − 1  2  �  Γ(s+2)  2(s+3)Γ  �  − s  2  �  Γ(s+3)  +  1  1+ s F2:1:1  1:1:1  \uf8ee  \uf8ef\uf8f0  −1− s  2  , −s  2 : 1  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' 1  2  1− s  2  , 1  2  : 1  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='−−  �������  −1,−1  \uf8f9  \uf8fa\uf8fb  (55)  Where F2:1:1  1:1:1(x, y) is the KdF function which converges for |�x| + |�y| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' So to evaluate it at (−1,−1), one needs  its analytic continuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' In the MATHEMATICA file Box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='nb, we provide a systematic derivation of the analytic  continuation for the same so that it converges at (−1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  For general Bn(s) we get the following MB-representation  Bn(s) =  1  Γ  �  − s  2  �  c1+i∞  �  c1−i∞  ···  cn−1+i∞  �  cn−1−i∞  �  n−1  �  p=1  dzp  2πi  � ��n−1  i=1 Γ(2zi +1)  �  Γ  �  s−2�n−1  j=1 z j +1  �  Γ  ��n−1  k=1 zk − s  2  �  ��n−1  l=1 Γ(2zl +2)  �  Γ  �  s−2�n−1  m=1 zm +2  �  (56)  Using the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (56) we obtain following representation for B4(s)  B4(s,α,β,γ) =  1  Γ  �  − s  2  �  c1+i∞  �  c1−i∞  c2+i∞  �  c2−i∞  c3+i∞  �  c3−i∞  Γ(−z1)Γ(2z1 +1)Γ(−z2)Γ(2z2 +1)Γ(−z3)Γ(2z3 +1)Γ(s−2z1 −2z2 −2z3 +1)  Γ(2z1 +2)Γ(2z2 +2)Γ(2z3 +2)Γ(s−2z1 −2z2 −2z3 +2)  ×Γ  �  − s  2 + z1 + z2 + z3  �  (α)z1(β)z2(γ)z3 dz1dz2dz3  (2πi)3  (57)  The above integral can be again evaluated readily using the MbConicHull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' For the case of B4(s), due to  the occurrence of a 3-variable hypergeometric function, the region of convergence analysis is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' In the OMOB  all the series which converges in the same region of convergence are kept together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' For 3 or more variables this  analysis becomes complicated and is not always straightforward [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Here the CHMB method plays an important role  in that it clubs the series converging in the same region of convergence together without prior knowledge of their  region of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The evaluation has been provided in the file Ising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='nb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2  ∆n(s)  We now move on to the evaluation of ∆n integrals (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Instead of directly doing the evaluation of the δn(s) integral,  we refer to [26], to exploit the relation between Bn(s) and ∆n(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' A few instances of the same are as follows:  ∆1(s) = 2  1  (s+1)(s+2)  (58)  ∆2(s) = 8  2  s  2 +1(s+3)+1  (s+2)(s+3)(s+4) +4B2(s)− 4(s+4)  s+2 B2(s+2)  (59)  ∆3(s) = 24  �  (s+5)  �  2  s  2 +3 −3  s  2 +2�  +1  �  (s+2)(s+4)(s+5)(s+6) + 24  s+2 B2(s+2)−  24(s+6)  (s+2)(s+4) B2(s+4)− 12(s+5)  s+2  B3(s+2)  (60)  + 4(s+6)(s+7)  (s+2)(s+4) B3(s+4)+8B3(s)  (61)  where B2(s) and B3(s) are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (50) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The results for ∆4 and ∆5 are provided in the appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='3  Jelium Potential  As an application of the evaluations done in the previous section, we refer to one more application of such evaluations,  the Jellium potential [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' It arises in the problem of electrostatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The problem concerns finding the electrostatic  potential energy of an electron (having charge -1) at the cube center, given an n-cube of uniformly charged jelly of  total charge +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' For the problem, usually one takes the radial potential at a distance r from the electron as Vn(r) as  follows  V1(r) := r −1/2,  V2(r) := log(2r),  Vn(r) := 2n−2 −  �1  r  �n−2  ,  n > 2  (62)  The n-th Jellium potential is defined as  Jn := 〈Vn(r)〉⃗r∈[−1/2,1/2]n  (63)  All the Jn can be written as a box integral up to an offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The final result is  Jn = 2n−2(1−Bn(2− n)),  n > 2  (64)  Using the result for Bn, J3 can be readily evaluated to:  J3 = π  2 +2−6tanh−1  � 1  �  3  �  (65)  6  Conclusion and Discussion  We show that using the MMOB [24] for the evaluation of improper integral with limits from 0 to ∞ combined with  tools to evaluate such MB integrals such as MbConicHull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl results in more efficient evaluation of these integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  This method is particularly helpful to evaluate the integrals when using OMOB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' one requires the use of ’regulators’  and further a proper limiting procedure to evaluate the integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The choice of these regulators is somewhat arbi-  trary, and at times more than one regulator has to be used, which further complicates the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' With these tools  at hand, we then re-evaluate the Ising integral, which had been already evaluated in [3] but with regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We  further make an attempt to evaluate the sought-after integral C5,k with all these techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We are, though, able to  evaluate a more general integral C5,k(α,β) which, when properly analytically continued, will give the result for C5,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  At present we are unable to do so with the techniques at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Though we believe that the result can be written as  a derivative of some multivariable hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Continuing further we evaluate the B3(s) and B4(s) and  give a general MB representation for Bn(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' For the case of B3(s), we use Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl to find the ACs of the hypergeo-  metric functions that appear in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' For B4(s), similar techniques would work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' It is important to note that  though the OMOB and the evaluation of MB representation will give essentially the same number of series, grouping  10  them in the same ROC is not an easy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' For the case of 3 or more variables, the problem of finding the ROC is  still a problem yet to be solved in an efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' This problem is essentially removed in the case of applying  the CHMB method, where such grouping is automatically done without prior knowledge of the ROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' As a byproduct  of these evaluations, we get the result for associated box integrals ∆n(s) and Jellium potential Jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We through these  evaluation also discover the relations between these integrals and multivariable hypergeometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  As a future direction, it would be interesting to modify the rules of the OMOB so that the final evaluation of  the bracket series doesn’t require regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' For the case of C5,k(α,β) evaluated in the present work, one can try  to find a way to evaluate the ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' One way towards this direction is to write the final result as a derivative of a  hypergeometric function and then find the ACs of it using Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' After finding the ACs, the derivative can be  taken to get the final result which converges in the appropriate region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We also note that a similar process can be  used to evaluate C6,k, which also gives a 2-fold MB integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Finally, it would be interesting to derive the result for  the various Box integrals Bn(s),∆n(s) and Jellium-potential Jn from the results given here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The result in the present  work matches numerically with those results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' it would still be interesting to see how they can be obtained from the  present work by using various reduction formulas of multivariable hypergeometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  7  Acknowledgements  TP would like to thank Souvik Bera for his help and his useful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  A  Ruby’s formula  Ruby’s formula is another interesting physical problem where the OMOB can still be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We provide an evaluation  of a general integral of which Ruby’s formula is a special case in this Appendix to highlight the application of the  OMOB when regulators are not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Ruby’s formula gives the solid angle subtended at a disk source by a coaxial  parallel-disk detector [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' It is given as follows  D = Rd  Rs  �∞  0  J1(kRd)J1(kRs) e−kd  k  dk  (66)  where Rd and Rs are the radii of the detector and the source, respectively, d is the distance between the source and  the detector, and J1(x) is the order one Bessel’s function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We now consider the generalization of  integral 66, as discussed in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' We will use the MOB to evaluate the integral and show that it reproduces the result,  along with two ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  S =  �∞  0  kl e−kd  N  �  j=1  Ja j(kR j)dk  (67)  we can again apply the method of brackets by using the series expansion of the functions  Ja j(kR j) = 1  2a j  ∞  �  n j=0  φn j  (kR j)2n j+a j  22n jΓ(a j + n j +1)  e−kd =  ∞  �  np=0  φnpknpdnp  putting the series expansion in the above integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' we get  S =  �∞  0  ∞  �  np=0  φnpknp+ldnp  N  �  j=1  1  2a j  ∞  �  n j=0  φn j  (kR j)2n j+1  22n jΓ(a j + n j +1)  dk  (68)  we can simplify the above by noting that  11  N  �  j=1  1  2a j  ∞  �  n j=1  φn j  (kR j)2n j+a j  22n jΓ(a j + n j +1)  =  ∞  �  n1=0  ···  ∞  �  nN=0  φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='···,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='Nk  �N  j=1(2n j+a j)  2  �N  j=1(2n j+a j)  ×  �N  j=1(R j)(2n j+a j)  �N  j=1 Γ(a j + n j +1)  putting above value in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (68) gives  S =  �∞  0  ∞  �  np=0  φnpk(np+l+�N  j=1(2n j+a j))dnp  ∞  �  n1=0  ···  ∞  �  nN=0  φ1,2,···,N  2  �N  j=1(2n j+a j)  ×  �N  j=1(R j)(2n j+a j)  �N  j=1 Γ(a j + n j +1)  dk  (69)  Using the method of brackets, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (69) can be written as  S =  ∞  �  n1=0  ···  ∞  �  nN=0  ∞  �  np=0  φ1,2,···,N,p〈(np + l +1+  N  �  j=1  (2n j + a j))〉  dnp  2  �N  j=1(2n j+a j)  ×  �N  j=1(R j)(2n j+a j)  �N  j=1Γ(a j + n j +1)  (70)  where φ1,2,···,N,p = φn1φn2 ···φnN φnp  The solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (70) are determined using the solution to the linear equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  np + l +1+  N  �  j=1  (2n j + a j) = 0  (71)  above equation has (N +1) variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' There are (N +1) different ways to write solutions to the above equation, taking  N free variables each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Out of (N+1) solutions, the solution with np as the dependent variable gives the Lauricella function of N variables,  as we will show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The rest of other solutions give the series representation that is the analytical continuation of the  earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Denoting the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (71) by n∗  i with ni being the dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  The solutions to equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (71) can be written as  n∗  p = −(l +1)−  N  �  j=1  (2n j + a j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='a = 1  n∗  i = −  (np + l +1)  2  −  N  �  j=1,i̸=j  (n j)−  N  �  j=1  �a j  2  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='a = 1  2  a is the coefficient of the dependent variable if the set of linear equations obtained from brackets are written in the  form an+ b = 0  where n is the dependent variable, and b includes all the free variables and the constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Denoting the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (70) by Si obtained by using n∗  i (i = 1,2,··· ,N, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  I) With np as the dependent variable  12  We write the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (70) as  Sp = 1  a  ∞  �  n1=0  ···  ∞  �  nN=0  φ1,2,···,NF(n1,n2,··· ,nN,n∗  p)Γ(−n∗  p)  (72)  where F(n1,n2,··· ,nN,np) =  dnp �N  j=1(R j)(2nj +a j)  2  �N  j=1(2nj +a j) �N  j=1 Γ(a j+n j+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Putting the values, we get  Sp =  ∞  �  n1=0  ···  ∞  �  nN=0  φ1,2,···,N  d−(l+1)−�N  j=1(2n j+a j) �N  j=1(R j)(2n j+a j)  2  �N  j=1(2n j+a j) �N  j=1Γ(a j + n j +1)  Γ  �  (l +1)+  N  �  j=1  (2n j + a j)  �  (73)  Using Legendre’s duplication formula  Γ  �  2  �l +1  2  +  N  �  j=1  �  n j + a j  2  ���  =  2  �  l+�N  j=1(2n j+a j)  �  Γ  �  l+1  2 +�N  j=1  �  n j +  a j  2  ��  Γ  �  l  2 +1+�N  j=1  �  n j +  a j  2  ��  �π  (74)  putting above value in equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (73) and simplifying gives  Sp =  ∞  �  n1=0  ···  ∞  �  nN=0  φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='···,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='d−(l+1)−�N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1(2n j+a j) �N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1(R j)(2n j+a j)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1(2n j+a j) �N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1 Γ(a j + n j +1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='Γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='l+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2 +�N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='n j +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='a j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='Γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='l  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2 +1+�N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='n j +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='a j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�π  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(75)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='this equation can be written in compact form as follow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='Sp = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�π  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='� 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�l� 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='Γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='� N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='a j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2 + l +1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='Γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='� N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='a j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2 + l  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
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+page_content='� N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�R j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�a j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='n1=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='···  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='nN=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1 n j �N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='� R j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�2n j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='a j +1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='n jΓ(n j +1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='��N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='a j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2 + l+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(�N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1 n j)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='��N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='a j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2 + l  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2 +1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(�N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1 n j)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j=1Γ(a j +1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(76)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(a)m is the Pochhammer symbol  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='which exactly matches the series representation obtained in [37] with ROC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='i=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='|R j| < d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='The above series corresponds to the Lauricella function of N variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Sp = 1  �π  � 2  d  �l� 1  d  ��  1  �N  j=1Γ(a j +1)  �  Γ  � N  �  j=1  a j  2 + l +1  2  �  Γ  � N  �  j=1  a j  2 + l  2 +1  � N  �  j=1  �R j  d  �a j  ×Fc  �� N  �  j=1  a j  2 + l +1  2  �  ,  � N  �  j=1  a j  2 + l  2 +1  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(1+ a1),··· ,(1+ aN);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='−  �R1  d  �2  ,··· ,−  � RN  d  �2�  (77)  where Fc in the above equation is the Lauricella function for N variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  II) With ni as the dependent variable  We write the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (70) as  Si = 1  a  ∞  �  n1=0  ···  ∞  �  nN=0  φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='···,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(i−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(i+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='···,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='pF(n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='n∗  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='nN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='np)Γ(−n∗  i )  (78)  putting the values,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' we get  Si = 1  2  ∞  �  n1=0  ···  ∞  �  ni−1=0  ∞  �  ni+1=0  ···  ∞  �  nN=0  ∞  �  np=0  φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='···,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(i−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(i+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='···,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='p  dnp  ��N  j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j̸=i(R j)(2n j+a j)�  �  2  �N  j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j̸=i(2n j+a j)���N  j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j̸=i Γ(a j + n j +1)  �  ×  �  1  �  Γ(ai −  (np+l+1)  2  −�N  n j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='i̸=j(n j)−�N  j=1  � a j  2  �  +1  �  ��  (Ri)−(np+l+1)−�N  j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='i̸=j(2n j)−�N  j=1 a j+ai�  ×  �  1  2−(np+l+1)−�N  j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='i̸=j(2n j)−�N  j=1 a j+ai  ��  Γ  � np + l +1  2  +  N  �  j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='j̸=i  (n j)+  N  �  j=1  �a j  2  ���  (79)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' (79) gives series representation for all values of i = 1,2,··· ,N and is the most general form of all the analytically  continued series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  B  ∆n Relations  ∆n can be expressed in terms of Bn as has already been shown in the subsection (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' the relations for ∆4 and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='∆5 are provided:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='∆4(s) = 64  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='3·2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2 +3 +2s+6 −3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='2 +4�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(s+7)+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(s+2)(s+4)(s+6)(s+7)(s+8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='96  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(s+2)(s+4) B2(s+4)−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='96(s+8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(s+2)(s+4)(s+6) B2(s+6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='(80)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
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+page_content='s+2 B3(s+2)−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='96(s+7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
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+page_content='∆5(s) = 1601+(9+ s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
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+page_content='26+s/2 +210+s −54+s/2 −2·35+s/2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
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+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
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+page_content='320(10+ s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
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+page_content='2+ s B4(2+ s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
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+page_content='(9+ s)(291+35s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
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+page_content='B5(8+ s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='MATHEMATICA files  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' we give a list of the MATHEMATICA files and packages that we provide,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' which contains the derivation of the  various results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Files Provided  Description  Ising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='nb  Contains the evaluation of the Ising integrals C3,k, C4,k, C5,k(α,β) and C6,k(α,β)  Box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='nb  Contains the evaluation of the Box integrals B3(s) and B4(s)  MbConicHull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl  Package required to evaluate multidimensional MB integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Used in the  evaluation of C5,k(α,β), C6,k(α,β), B3(s) and B4(s)  MultivariateResidues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='m  Used by the package MbConicHull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl internally  Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl  Package required for finding the ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Used for the case B3(s)  ROC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='wl  Package required for finding the region of convergence of the 2-variable hypergeometric series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Table 2:  References  [1] Izrail Solomonovich Gradshteyn and Iosif Moiseevich Ryzhik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Table of integrals, series, and products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Academic  press, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [2] Ivan Gonzalez and Victor H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Moll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Definite integrals by the method of brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Advances in Applied Mathemat-  ics, 45(1):50–73, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [3] Ivan Gonzalez, Victor H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Moll, and Armin Straub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The Method of brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Part 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Examples and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  4 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [4] Ivan Gonzalez, Karen Kohl, Lin Jiu, and Victor H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Moll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' An extension of the method of brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Open  Mathematics, 15(1):1181–1211, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [5] Ivan Gonzalez, Lin Jiu, and Victor H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Moll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' An extension of the method of brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' part 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Open Mathematics,  18(1):983–995, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [6] Ivan Gonzalez, Igor Kondrashuk, Victor H Moll, and Luis M Recabarren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Mellin–barnes integrals and the  method of brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' The European Physical Journal C, 82(1):28, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [7] Ivan Gonzalez, Igor Kondrashuk, Victor H Moll, and Alfredo Vega.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Analytic expressions for debye functions and  the heat capacity of a solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Mathematics, 10(10):1745, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [8] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Ananthanarayan, Sumit Banik, Sudeepan Datta, and Tanay Pathak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Quadratic and quartic integrals using  the method of brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Scientia, 29:45–59, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [9] Hari M Srivastava and Per Wennerberg Karlsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Multiple Gaussian hypergeometric series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Horwood, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [10] Harold Exton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Multiple hypergeometric functions and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Ellis Horwood, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [11] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Ananthanarayan, Sumit Banik, Samuel Friot, and Shayan Ghosh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
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+page_content=' Square lattice ising model susceptibility: series expansion  method and differential equation for χ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Journal of Physics A: Mathematical and General, 38(9):1875, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [33] David H Bailey, Jonathan M Borwein, David Broadhurst, and Wadim Zudilin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Experimental mathematics and  mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Math, 517:41–58, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [34] David H Bailey and Jonathan M Borwein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  High-precision numerical integration: Progress and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Journal of Symbolic Computation, 46(7):741–754, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [35] Per OM Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Integration of the partial differential equations for the hypergeometric functions f 1 and fd of  two and more variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Journal of Mathematical Physics, 5(3):420–430, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [36] Lawrence Ruby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Further comments on the geometrical efficiency of a parallel-disk source and detector system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and  Associated Equipment, 337(2):531–533, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  [37] Samuel Friot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' On Ruby’s solid angle formula and some of its generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content=' A, 773:150–  153, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}
+page_content='  16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'}