The method of brackets. Part 2: examples and applications

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1 Contemporary Mathematic The method of bracket Part : example and application Ivan Gonzalez, Victor H Moll, and Armin Straub Abtract A new heuritic method for the evaluation of definite integral i preented Thi method of bracket ha it origin in method developed for the evaluation of Feynman diagram The operational rule are decribed and the method i illutrated with everal example The method of bracket reduce the evaluation of a large cla of definite integral to the olution of a linear ytem of equation Introduction The method of bracket preented here provide a method for the evaluation of a large cla of definite integral The idea were originally preented in [6] in the context of integral ariing from Feynman diagram A complete decription of the operational rule of the method together with a variety of example wa firt dicued in [5] The method i quite imple to work with and many of the entrie from the claical table of integral [7] can be derived uing thi method The baic idea i to introduce the formal ymbol a, called a bracket, which repreent the divergent integral x a dx The formal rule for operating with thee bracket are decribed in Section and their jutification epecially of the heuritic Rule 3 i work-in-progre In particular, convergence iue are ignored at the moment Roughly, each integral generate a linear ytem of equation and for each choice of free variable the method yield a erie with the free variable a ummation indice A heuritic rule tate that thoe converging in a common region give the deired evaluation Section 3 illutrate the method by evaluating the Laplace tranform of the Beel function J ν x In thi example, the two reulting erie converge in different region and are analytic continuation of each other Thi i a general phenomenon which i ued in Section 5 to produce an explicit analytic continuation of the hypergeometric function q+ F q x Section 4 preent the evaluation of a family of integral C n appearing in Statitical Mechanic Thee were introduced in [4] a a Mathematic Subject Claification Primary 33C5, Secondary 33C67, 8T8 Key word and phrae Definite integral, hypergeometric function, Feynman diagram c copyright holder

2 IVAN GONZALEZ, VICTOR H MOLL, AND ARMIN STRAUB toy model and their phyical interpretation wa dicovered later The method of bracket i employed here to evaluate the firt four value, the only known cae an expreion for the next value C 5 in term of a double hypergeometric erie i poible but i not given here The lat ection employ the method of bracket to reolve a Feynman diagram The method of bracket The method of bracket dicued in thi paper i baed on the aignment of the formal ymbol a to the divergent integral Example If f i given by the formal power erie fx = a n x αn+β, n= then the improper integral of f over the poitive real axi i formally written a the bracket erie fxdx = a n αn+β n Here, and in the equel, n i ued a a horthand for n= Formal rule for operating with bracket are decribed next In particular, Rule decribe how to evaluate a bracket erie uch a the one appearing in To thi end, it i ueful to introduce the ymbol φ n = n n+, which i called the indicator of n Example The gamma function ha the bracket expanion 3 a = x a e x dx = n φ n n+a Rule The bracket expanion 4 a +a + +a r α = φ m,,m r a m a mr r m,,m r α+m + +m r α hold Here φ m,,m r i a horthand notation for the product φ m φ mr If there i no poibility of confuion thi will be further abridged a φ {m} The notation i to be undertood likewie {m} Rule A erie of bracket i aigned a value according to 5 φ n fn an+b = a fn n, n where n i the olution of the equation an+b = Oberve that thi might reult in the replacing of the index n, initially a nonnegative integer, by a complex number n Similarly, a higher dimenional bracket erie, that i, φ {n} fn,,n r a n + a r n r +c a r n + a rr n r +c r {n}

3 METHOD OF BRACKETS 3 i aigned the value 6 deta fn,,n r n n r, where A i the matrix of coefficient a ij and n i i the olution of the linear ytem obtained by the vanihing of the bracket The value i not defined if the matrix A i not invertible Rule 3 In the cae where a higher dimenional erie ha more ummation indice than bracket, the appropriate number of free variable i choen among the indice For each uch choice, Rule yield a erie Thoe converging in a common region are added to evaluate the deired integral 3 An example from Gradhteyn and Ryzhik The econd author i involved in a long term project of providing proof of all the entrie from the claical table of integral by Gradhteyn and Ryzhik [7] The proof can be found at: 3 In thi ection the method of bracket i illutrated to find which i entry 663 of [7] Here 3 J ν x = x ν e αx J ν βxdx = βν ν + πα +β ν+/ k= k x/ k+ν k!k +ν + i the Beel function of order ν To thi end, the integrand i expanded a e αx J ν βx = φ n αx n βx 33 φ k+ν k k +ν + n k = k,n o a to obtain the bracket erie 34 e αx J ν βxdx = k,n φ k,n α n β k+ν k +ν + xn+k+ν, α n β φ k+ν k,n n+k +ν + k +ν + The evaluation of thi double um by the method of bracket produce two erie correponding to uing either k or n a the free variable when applying Rule The index k i free Chooing k a the free variable when applying Rule to 34, yield n = k ν and thu the reulting erie 35 k α k ν β φ k+ν k k +ν + k +ν + ν + = α ν β + νν ν + F β α

4 4 IVAN GONZALEZ, VICTOR H MOLL, AND ARMIN STRAUB The right-hand ide employ the uual notation for the hypergeometric function a,,a p 36 pf q b,,b q x a = n a p n x n b n b q n n! where α n = α+n α i the Pochhammer ymbol Note that the F in 35 converge provided β < α In thi cae, the tandard identity F a x = x a together with the duplication formula for the function how that the erie in 35 i indeed equal to the right-hand ide of 3 The index n i free In thi econd cae, the linear ytem in Rule ha determinant and yield k = n/ ν / Thi give 37 n n= α n β φ n ν n n/+ν +/ n/+/ Thi erie now converge provided that β > α in which cae it again um to the right-hand ide of 3 Note Thi i the typical behavior of the method of bracket The different choice of indice a free variable give repreentation of the olution valid in different region Each of thee i an analytic continuation of the other one 4 Integral of the Iing cla In thi ection the method of bracket i ued to dicu the integral 4 C n = 4 n! du n du n j= u u j +/u j u n Thi family wa introduced in [4] a a caricature of the Iing uceptibility integral 4 D n = 4 ui u j du n! u i<j i +u j n du n j= u u j +/u j u n Actually, the integral C n appear naturally in the analyi of certain amplitude tranform [] The firt few value are given by 43 C =, C =, C 3 = L 3, C 4 = 7 ζ3 Here, L D i the Dirichlet L-function In thi cae, 44 L 3 = n= 3n+ 3n+ No analytic expreion for C n i known for n 5 Similarly, 45 D =, D = 3, D 3 = 8+ 4π 3 7L 3, D 4 = 4π ζ3 are given in [4] High preciion numerical evaluation and PSLQ experiment have further produced the conjecture D 5 = 4 984Li π4 74ζ3 7ζ3ln+4π ln 6 3 π π ln+88ln ln 4ln

5 METHOD OF BRACKETS 5 The integral C n i the pecial cae k = of the family 47 C n,k = 4 n! du n k+ du n j= u u j +/u j u n that alo give the moment of power of the Beel function K via 48 C n,k = n k+ c n,k := n k+ n!k! n!k! The value 49 c,k = k k + a well a the recurion π, c,k = 4 t k K n tdt 3 k+ k +, 4 k + 4 c 3,k 5k +k +c 3,k+ +9c 3,k+4 = with initial data 4 c 3, = 3α 3π, c 3, = 3 4 L 3, c 3, = α 96π 4π5 9α, c 3,3 = L 3 3, where α = /3 6 3 are given in [] and [3] The evaluation of thee integral preented in the literature uually begin with the introduction of pherical coordinate Thi reduce the dimenion of C n by two and immediately give the value of C and C The evaluation of C 3 i reduced to the logarithmic integral 4 C 3 = 3 ln+xdx x +x+ It value i obtained by the change of variable x t followed by an expanion of the integrand A ytematic dicuion of thee type of logarithmic integral i provided in [9] The value of C 4 i obtained via the double integral repreentation 43 C 4 = 6 Moreover, the limiting behavior ln+x+y +x+y+/x+/y 44 lim n C n = e γ dxdy x y wa etablihed in [4] In thi ection the method of bracket i ued to obtain the expreion for C, C 3, and C 4 decribed above An advantage of thi method i that it ytematically give an analytic expreion for thee integral When applied to C 5, the method produce a double erie repreentation which i not dicued here 4 Evaluation of C,k The number C,k are given by 45 C,k = dxdy xy x+/x+y +/y k+ A direct application of the method of bracket, by applying Rule to the integrand a in 45, reult in a bracket expanion involving a 4-fold um and 3 bracket Rule and 3 tranlate thi into a collection of erie with 4 3 =

6 6 IVAN GONZALEZ, VICTOR H MOLL, AND ARMIN STRAUB ummation indice However, it i generally deirable to minimize the final number of ummation by reducing the number of um and increaing the number of bracket In thi example thi i achieved by writing C,k = = xy k dxdy x y +y +xy +x k+ xy k dxdy xy[x+y]+[x+y] k+ In the evaluation of thee expreion, the term x+y mut be expanded at the lat tep The method of bracket now yield xy[x+y]+[x+y] k+ = φ n,n x n y n x+y k ++n n+n +n, k + n,n and the expanion of the term x+y give x+y = φ n3,n n n 4 x n3 y n n4 n +n 3 +n 4 n n n 3,n 4 Replacing in the integral produce the bracket expanion C,k = {n} k ++n +n n n +n 3 +n 4 φ {n} k + n n k ++n +n 3 k ++n +n 4 The value of thi formal um i now obtained by olving the linear ytem k ++ n +n =, n n +n 3 +n 4 =, k++n +n 3 =, and k++n +n 4 = coming from the vanihing of bracket Thi ytem ha determinant and it unique olution i n = n = n 3 = n 4 = k+ It follow that 46 C,k = n n n 3 n 4 k + n n k+ 4 = k + Note that, upon employing Legendre duplication formula for the function, thi evaluationiequivalentto49 Inparticular,thiconfirmthevalueC = C, = in Remark 4 The evaluation C,k α,β = = k++α+β x α y β dxdy x+/x+y +/y k+ k+ α β k + that generalize C,k i obtained a a bonu Similarly, 48 J r, α,β = = r+α+β x α y β dxdy x+y r xy + +r α β r Note that C,k α,β = J k+,k+ α+k +,β +k + k++α β r+α β k+ α+β r α+β

7 METHOD OF BRACKETS 7 Remark 4 TheIinguceptibilityintegralD, ee4, iobtaineddirectly from the expreion for J r, given above Indeed, 49 D = x xy +y xydxdy x+y 4 xy + = J 4, 4, J 4, 3,3+J 4,,4 = 3 Thi agree with 45 Thi technique alo yield the generalization x y x α y β dxdy 4 D α,β = x+y x+/x+y +/y with limiting cae D α,α = 3 4 = b ab+a+b a π coαπ coβπ απ inαπ 4 Evaluation of C 3,k Next, conider the integral C 3,k = 3 = 3 dxdydz xyz x+/x+y +/y +z +/z k+ xyz k dxdydz xyzx+y+zx+y+xyz +xy k+ The econd form of the integrand i motivated by the deire to to minimize the number of um and to maximize the number of bracket in the expanion The denominator i now expanded a φ {n} xy n+n3+n4 z n+n+n3 x+y k ++n n+n +n +n 3 +n 4, k + {n} and further expanding x+y n+n a x+y n+n = φ n5,n 6 x n5 y n n6 n +n 5 +n 6 n n n 5,n 6 produce a complete bracket expanion of the integrand of C 3,k Integration then yield C 3,k = n n +n 5 +n 6 4 φ {n} 3k! n n {n} k ++n +n +n 3 +n 4 k ++n +n 3 +n 4 +n 5 k ++n +n 3 +n 4 +n 6 k ++n +n +n 3 Thi expreion i regularized by replacing the bracket k++n +n +n 3 with k ++n +n +n 3 +ǫ with the intent of letting ǫ Thi correpond to multiplying the initial integrand with z ǫ ; however, note that many other regularization are poible and eventually lead to Theorem 43 It will become clear hortly, ee 44, why regularizing i neceary The method of bracket now give a et of erie expanion obtained by the vanihing of the five bracket in 4 The olution of the correponding linear ytem which ha determinant leave one free index and produce the integral a a erie in thi variable Of the ix poible

8 8 IVAN GONZALEZ, VICTOR H MOLL, AND ARMIN STRAUB free indice, only n 3 and n 4 produce convergent erie more pecifically, for each free index one obtain a hypergeometric erie 3 F time an expreion free of the index; for the indice n 3,n 4 the argument of thi 3 F i 4 while otherwie it i 4 The heuritic Rule 3 tate that their um yield the value of the integral: 43 C 3,k = 3 lim n f k,n ǫ+f k,n ǫ ǫ k! n! n= where 44 f k,n ǫ = 4 n ǫ n+ k++ǫ n+k ++ǫ Oberve that the term f k,n ǫ are contributed by the index n 3 while the term f k,n ǫ come from the index n 4 At ǫ =, each of them ha a imple pole Conequently, the even combination f k,n ǫ+f k,n ǫ ha no pole at ǫ = Uing the expanion 45 x+ǫ = x+ψxǫ+oǫ, for x,,,, a well a 46 n+ǫ = n n! for n =,,,, provide the next reult ǫ +ψn+ +Oǫ, Theorem 43 The integral C 3,k are given by C 3,k = n+ k+ 4 ψn+ ψ n+ k+ 3k! n! n+k + +ψn+k + n= In particular, for k = 47 C 3 = n! 3 n+! ψn+ ψn+ n= The evaluation of thi um uing Mathematica 7 yield a large collection of pecial value of poly-logarithm After implification, it yield C 3 = L 3 a in 43 Remark 44 An extenion of Theorem 43 i preented next: 48 C 3,k α,β,γ = for γ =, i given by k! n! n= n+ k+±α±β n+k + ψn+ ψ x α y β z γ dxdydz x+/x+y +/y +z +/z k+, n+ k+±α±β +ψn+k + where the notation n+ k+±α±β = n+ k++α+β n+ k++α β a well a ψn+ k+±α±β = ψn+ k++α+β +ψn+ k++α β + i employed Similar expreion can be given for other integral value of γ In the cae where γ i not integral, C 3,k α,β,γ can be written a a um of two 3 F with factor The ymmetry of C 3,k α,β,γ in α,β,γ, how that thi can be done if at leat one of thee argument i nonintegral

9 METHOD OF BRACKETS 9 43 Evaluation of C 4 The lat example dicued here i C 4 = 6 dxdydzdw xyzwx+/x+y +/y +z +/z +w +/w To minimize the number of um and to maximize the number of bracket thi i rewritten a 6 x +ǫ y +ǫ z +ǫ w +ǫ dxdydzdw [Axyzwx+y+zwx+y+xyzwz +w+xyz +w] with the intent of letting ǫ and A A in the cae of C 3,k, the regulator parameter ǫ i introduced to cure the divergence of the reulting expreion Similarly, the parameter A i employed to divide the reulting um into convergence group according to the heuritic Rule 3 The denominator expand a φ {n} A n x n+n3+n4 y n+n3+n4 z n+n+n3 w n+n+n3 {n} x+y n+n z +w n3+n4 +n +n +n 3 +n 4 A before, x+y n+n = φ n5,n 6 x n5 y n n6 n +n 5 +n 6 n n n 5,n 6 and z +w n3+n4 = φ n7,n 8 z n7 w n n8 3 n 4 +n 7 +n 8 n n 3 n 4 7,n 8 Thee expanion of the integrand yield the bracket erie 49 6 φ {n} A n +n +n +n 3 +n 4 {n} n n +n 5 +n 6 n 3 n 4 +n 7 +n 8 n n n 3 n 4 +ǫ+n +n 3 +n 4 +n 5 +ǫ+n +n 3 +n 4 +n 6 +ǫ+n +n +n 3 +n 7 +ǫ+n +n +n 3 +n 8 The evaluation of thi bracket erie by Rule and 3 yield hypergeometric erie with argument A n, n, n 5, or n 6 choen a the free index and /A n 3, n 4, n 7, or n 8 choen a the free index Either combination produce an expreion for the integral C 4 Taking thoe with argument A the indice n 5 and n 6 yield the ame erie; however, it i only taken into account once give 43 A A ǫ ǫ ǫ ǫ +ǫ F + A ǫ ǫ F +ǫ, 3 +ǫ A ǫ, A 3 ǫ F, 3 A

10 IVAN GONZALEZ, VICTOR H MOLL, AND ARMIN STRAUB A ǫ, the limiting value i + A 43 4 ln A ln A + [ 3 Li 3 A Li 3 ] A A lna 6 A Finally, the value of C 4 i obtained by taking A : [ Li A Li ] A 43 C 4 = 3 [Li 3 Li 3 ] = 7 ζ3 Thi agree with 43 5 Analytic continuation of hypergeometric function The hypergeometric function p F q, defined by the erie a,,a p 5 pf q x = p F q b,,b q x a n a p n x n = b n b q n n!, converge for all x C if p < q+ and for x < if p = q+ In the remaining cae, p > q+, theerie diverge for x Theanalytic continuation of theerie q+ F q ha been recently conidered in [, ] In thi ection a bracket repreentation of the hypergeometric erie i obtained and then employed to produce it analytic extenion Theorem 5 The bracket repreentation of the hypergeometric function i given by pf q x = [ φ n,{t},{} q x ] n p a j +n+t j q b k n+ k n a j= j b k k= t,,t p,, q n= Proof Thi follow from 5 and the repreentation 5 a j n = a j +n a j a well a 53 = a j τ aj+n e τ dτ = t j φ tj a j +n+t j a j = n b k n = n b k n+ k φ k b k n b k b k k for the Pochhammer ymbol The bracket expreion for the hypergeometric function given in Theorem 5 contain p + q bracket and p + q + indice n, t j and k Thi lead to a full rank ytem 54 a j +n+t j = for j p b k n+ k = for k q of linear equation of ize p+q+ p+q and determinant For each choice of an index a a free variable the method of bracket yield a one-dimenional erie for the integral

11 METHOD OF BRACKETS Serie with n a a free variable Solving 54 yield t j = a j n and k = b k+n with j p and k q Rule yield [ q x] n p n+a j q n+ b k a n a p n x n = n! a j b k b n b q n n! n= j= k= Thi i the original erie repreentation 5 of the hypergeometric function In particular, in the cae q = p, thi erie converge for x < Serie with t i a a free variable Fix an index i in the range i p and olve 54 to get n = a i t i, a well a t j = t i a j +a i for j p, j i, and k = b k a i t i for k q The method of bracket then produce the erie [ φ ti q x ] ti ai t i +a i a t i i j i a j a i t i a j which may be rewritten a a x ai j a i b k 55 a j b k a i j i k ai,{ b k +a i } k q q+ F p { a j +a i } j p,j i n= k b k +a i +t i b k p+q Recall that the initial hypergeometric erie p F q x converge for ome x if and only if p q+ Hence, auming that p q+, oberve that the hypergeometric erie 55 converge for ome x if and only if p = q + Serie with i a a free variable Proceeding a in the previou cae and chooing i in the range i q and then i a the free index, give [ p+q x ] b i b i a j b i b k 56 b i b j i a j b k k i {a j + b i } j p p F q b i,{ b k +b i } k q,k i x Summary Aume p = q+ and um up the erie coming from the method of bracket converging in the common region x > Rule 3 give the analytic continuation 57 q+ q+f q x = x ai i= j i a j a i a j k b k b k a i x ai,{ b k +a i } k q q+ F q { a j +a i } j q+,j i x for the erie 5 On the other hand, the q+ function coming from chooing n or i, i q, a the free variable form linearly independent olution to the hypergeometric differential equation 58 q+ j= x d dx +a j y = q k= x d dx +b k y

12 IVAN GONZALEZ, VICTOR H MOLL, AND ARMIN STRAUB inaneighborhoodofx = Likewie, theq+function55comingfromchooing t i, i q +, a the free variable form linearly independent olution to 58 in a neighborhood of x = Example 5 For intance, if p = and q = then a,b F c x = x ab ac a, c+a bc a F 59 b+a x + x ba bc b, c+b ac b F a+b x Thi i entry 93 of [7] On the other hand, the two function a,b 5 F a+ c,b+ c c x, x c F c x form a bai of the olution to the econd-order hypergeometric differential equation 5 x d dx +a x d dx +b y = x d dx +c y in a neighborhood of x = 6 Feynman diagram application In Quantum Field Theory the permanent contrat between experimental meaurement and theoretical model ha been poible due to the development of novel and powerful analytical and numerical technique in perturbative calculation The fundamental problem that arie in perturbation theory i the actual calculation of the loop integral aociated to the Feynman diagram, whoe olution i pecially difficult ince thee integral contain in general both ultraviolet UV and infrared IR divergence Uing the dimenional regularization cheme, which extend the dimenionality of pace-time by adding a fractional piece D = 4 ǫ, it i poible to know the behavior of uch divergence in term of Laurent expanion with repect to the dimenional regulator ǫ when it tend to zero A an illutration of the ue of method of bracket, the Feynman diagram 6 P a a 3 P a P 3 conidered in [] i reolved In thi diagram the propagator or internal line aociated to the index a ha ma m and the other parameter are P = P3 = and P = P +P 3 = The D-dimenional repreentation in Minkowki pace i given by d D q 6 G = iπ D/ [P +q m ] a [P 3 q ] a [q ] a3

13 METHOD OF BRACKETS 3 In order to evaluate thi integral, the Schwinger parametrization of 6 i conidered ee [8] for detail Thi i given by 63 G = D/ 3 j= a j H with H defined by 64 H = x a x a x a3 3 exp x m exp xx x +x +x 3 dx x +x +x 3 D/ dx dx 3 To apply the method of bracket the exponential term are expanded a exp x m x x exp = n x +n φ n,n x +x +x n m n n x n 3 x n +x +x 3 n,,n and then 64 i tranformed into 65 n,n φ n,n m n n x a+n+n x a+n x a3 3 dx dx dx 3 x +x +x 3 D/+n Further expanding = x +x +x 3 D/+n n 3,n 4,n 5 φ n3,n 4,n 5 x n3 xn4 xn5 3 D +n +n 3 +n 4 +n 5 D +n and replacing into 65 and ubtituting the reulting integral by the correponding bracket yield H = D φ {n} n m n n +n +n 3 +n 4 +n 5 66 D {n} +n a +n +n +n 3 a +n +n 4 a 3 +n 5 Thi bracket erie i now evaluated employing Rule and 3 Poible choice for free variable are n, n, and n 4 The erie aociated to n converge for m <, wherea the erie aociated to n,n 4 converge for m < The following two repreentation for G follow from here Theorem 6 In the region m <, a +a +a 3 D 67 H = η,a F D m with η defined by D a a 3 η = a m D a a a3 a 3 a +a +a 3 D D,

14 4 IVAN GONZALEZ, VICTOR H MOLL, AND ARMIN STRAUB 68 Theorem 6 In the region m <, a +a +a 3 D H = η,+a +a +a 3 D F +a +a 3 D +a D +η 4,a F m a a 3 + D with η, η 4 defined by m D a a 3 D a a 3 η = D a a a3a 3 a +a +a 3 D D a a a 3 D a a 3 η 4 = a a m D a a3 a 3 a +a 3 D D a Thee two olution are now pecialized to a = a = a 3 = Thi ituation i pecially relevant, ince when an arbitrary Feynman diagram i computed, the indice aociated to the propagator are normally Then, with D = 4 ǫ, the equation 67 and 68 take the form 69 H = m ǫ ǫ F +ǫ, ǫ for m <, a well a 6 H = ǫ ǫ +ǫ ǫ m m ǫ m ǫǫ ǫ, ǫ F ǫ m for m < Oberve that thee repreentation both have a pole at ǫ = of firt order for the econd repreentation, each of the ummand ha a pole of econd order which cancel each other 7 Concluion and future work The method of bracket provide a very effective procedure to evaluate definite integral over the interval [, The method i baed on a heuritic lit of rule on the bracket erie aociated to uch integral In particular, a variety of example that illutrate the power of thi method ha been provided A rigorou validation of thee rule a well a a ytematic tudy of integral from Feynman diagram i in progre Acknowledgment The firt author wa partially funded by Fondecyt Chile, Grant number 389 The work of the econd author wa partially funded by NSF-DMS 7567 The lat author wa funded by thi lat grant a a graduate tudent Reference [] D H Bailey, J M Borwein, D M Broadhurt, and L Glaer Elliptic integral repreentation of Beel moment J Phy A: Math Theor, 4:53 53, 8 [] E E Boo and A I Davydychev A method of evaluating maive Feynman integral Theor Math Phy, 89:5 63, 99 [3] J M Borwein and B Salvy A proof of a recurion for Beel moment Experimental Mathematic, 7:3 3, 8,

15 METHOD OF BRACKETS 5 [4] J M Borwein D H Bailey and R E Crandall Integral of the Iing cla Jour Phy A, 39:7 3, 6 [5] I Gonzalez and V Moll Definite integral by the method of bracket Part Adv Appl Math, To appear, [6] I Gonzalez and I Schmidt Optimized negative dimenional integration method NDIM and multiloop Feynman diagram calculation Nuclear Phyic B, 769:4 73, 7 [7] I S Gradhteyn and I M Ryzhik Table of Integral, Serie, and Product Edited by A Jeffrey and D Zwillinger Academic Pre, New York, 7th edition, 7 [8] C Itzykon and J B Zuber Quantum Field Theory World Scientific, Singapore, nd edition, 993 [9] L Medina and V Moll A cla of logarithmic integral Ramanujan Journal, :9 6, 9 [] J Palmer and C Tracy Two-dimenional Iing correlation: Convergence of the caling limit Adv Appl Math, :39 388, 98 [] S L Skorokhodov Method of analytic continuation of the generalized hypergeometric function pf p a,,a p;b,,b p ;z Comp Math and Math Phyic, 44: 3, 4 [] S L Skorokhodov Symbolic tranformation in the problem of analytic continuation of the hypergeometric function pf p z to the neighborhood of the point z = in the logarithmic cae Programming and Computer Software, 3:5 56, 4 Departmento de Fiica y Centro de Etudio Subatomico, Univeridad Santa Maria, Valparaio, Chile addre: ivangonzalez@umcl Department of Mathematic, Tulane Univerity, New Orlean, LA 78 addre: vhm@mathtulaneedu Department of Mathematic, Tulane Univerity, New Orlean, LA 78 addre: atraub@mathtulaneedu

POCHHAMMER SYMBOL WITH NEGATIVE INDICES. A NEW RULE FOR THE METHOD OF BRACKETS.

POCHHAMMER SYMBOL WITH NEGATIVE INDICES A NEW RULE FOR THE METHOD OF BRACKETS IVAN GONZALEZ, LIN JIU, AND VICTOR H MOLL Abstract The method of brackets is a method of integration based upon a small number