Advances in the theory of box integrals

Transcription

1 Advances in the theory of box integrals D.H. Bailey J.M. Borwein R.E. Crandall December, 9 Abstract. Box integrals expectations r s or r q s over the unit n-cube have over three decades been occasionally given closed forms for isolated n, s. By employing experimental mathematics together with a new, global analytic strategy, we prove that for each of n =,, 3, dimensions the box integrals are for any integer s hypergeometrically closed hyperclosed ) in an explicit sense we clarify herein. For n = 5 dimensions, such a complete hyperclosure proof is blocked by a single, unresolved integral we call K 5 although we do prove that all but a finite set of n = 5) cases enjoy hyperclosure. We supply a compendium of exemplary closed forms that arise naturally from the theory. Lawrence Berkeley National Laboratory, Berkeley, CA 97, dhbailey@lbl.gov. Supported in part by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC- 5CH3. School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 38, Australia jonathan.borwein@newcastle.edu.au and Faculty of Computer Science, Dalhousie University, Halifax, NS, B3H W5, Canada, jborwein@cs.dal.ca. Supported in part by ARC, NSERC and the Canada Research Chair Programme. Center for Advanced Computation, Reed College, Portland OR, crandall@reed.edu.

2 Preliminaries We define box integrals for positive-integer dimension n as expectations of r s, r q s with the relevant vectors chosen randomly, independently, equidistributed over the unit n-cube [3]: B n s) := r s D r r [,] n = n s) := = ) r + + rn s/ dr dr n, ) r, q [,] n r q s D r D q r q ) + + r n q n ) ) s/ dr dr n dq dq n. ) Here, D r denotes simply the volume element dr dr. As explained in a previous treatment [5], there are physical interpretations:. B n ) is the expected distance of a random point from the origin or from any fixed vertex) of the n-cube;. n ) is the expected distance between two random points of the n-cube; 3. B n n + ) is the expected electrostatic potential in an n-cube whose origin has a unit charge; 3. n n + ) is the expected electrostatic energy between two points in a uniform cube of charged jellium. Note that the definitions show immediately that both n m) and B n m) are rational when m, n are natural numbers. A pivotal, original treatment on box integrals is the 976 work of Anderssen et al, []. There have been interesting modern treatments of the B n and related integrals, as in [7], [, p.8], [7] and [5]. Related material is also to be found in [6, 6]. There are other similar entities such as the expected distance between points on distinct sides of a cube or hypercube investigated in [,.7] and [7]. We remark that B 3 ) is also known as the Robbins constant, after []. It is worth noting that the methods of the present paper actually have practical application for example in biology, of all places. In the work [] these new box-integral Not to be confused with box integrals of particle physics. We refer to box integrals but use the term n-cube for the domain hypercube. 3 Such statements presume that electrostatic potential in n dimensions is V r) = /r n, and say log r for n = ; the main issue being that negative powers of r can also have physical meaning.

3 evaluations are being used as one measure of how random? is a point cloud. That is, one test of whether brain synapses are sufficiently randomly distributed involves comparison of empirical values r q s, where r, q run over synapse positions, against our present theoretical tabulations of 3 s). Quadrature formulae for all complex powers As in our previous treatment [5], we define two key functions as bu) := e u x π erfu) dx =, 3) u du) := e u x y) dx dy = + e u + π u erfu) u. ) These functions may be used as integration kernels, in the following way. Writing R s = Γ s/) t s/ e tr dt, 5) valid for Rs) <, we can integrate formally over R being a radius, or a separation) in a relevant region, to obtain B n s) = n s) = Γ s/) Γ s/) u s b n u) du, 6) u s d n u) du, 7) both representations being valid for a reduced range of complex s, namely the range Rs) n, ). This domain of convergence for the integrals can be inferred from the large-u asymptotic behaviors, see e.g., [], du) bu) = e u + O u), 8) π u u + O e u), 9) so that having integrand factors b n, d n respectively allows integral convergence for the stated range Rs) n, ). Throughout n will denote a natural number. What can we say about analytic continuation in s? We can extend such previous results to all complex s, by paying deeper attention to integrand structure. First, we can extend the region for negative Rs) arbitrarily, by finding superpositions of terms bku) or dku), for integers k, such that the n-th powers of the asymptotic forms 8, 9) are genuinely of exponential decay. 3

4 Second, for positive Rs) and a positive integer K, we may employ the integral identity Dρ, K) := t ρ e tu)k du = Γ ρ) K j= ) K ) j j ρ. ) j from which it is straightforward to derive integral representations for all s in the open right half-plane except for the positive even integers. Theorem [Quadrature formulae for general complex s] In the region with Rs) < we have B n s) = n+s Γ s/) u s b n u) n b n u)) du. ) For n s) in the same s-region, let A := and consider the unique solution A,...,A n+ ) to the linear system Then n s) = = n+ k= A k, q =,,,..., n. ) kq n+ ) n+ ) A k k n+s u s A k k n d n ku) du. 3) Γ s/) k= Next, consider the region Rs), K) with K a positive integer and s not an even integer. We have B n s) = n s) = u s du Ds/, K) u s du Ds/, K) j= k= K K ) ) j b n u ) j, j ) K K ) ) j d n u ) j. j 5) Remark: As for nonnegative even integers s not covered in the above theorem, the definitions, ) immediately yield rational values for B n, D n respectively upon symbolic integration. It is interesting that even though the goal of the above development is to provide practical quadrature formulae, we already have a byproduct: Corollary [Pole structure of the box integrals] B n s), for positive integer n, has precisely one pole, namely at s = n. n s), on the other hand, always has precisely n + ) poles, located at s = n, n +,..., n. j=

5 Proof: Clearly, by the original definitions ) and ), B n s), n s) are both finite for nonnegative Rs) so it suffices to look at negative Rs) in any case. For B n s), relation ) has a pole factor at s = n. As for n s), the solution A k ) to the system of equations is unique since the matrix with entries ) k /k + ) j for k, j n is nonsingular with determinant ) n/ n i= i! n+)! see sequence A9 in Sloane s Online Encyclopedia of Integer Sequences). Poles of n force the prefactor n+ k= A kk n+s to vanish exactly when n + s [ n, ], thus giving a total of n + ) poles on the negative real s-axis. It remains to check that n+ k= A kk n+s has no other zeros. This runs as follows. Consider the more general exponential sum Φ N t) := N α j βj t = j= for arbitrary real constants α j and positive β j. If Φ N t) = has more than N solutions then Rolle s theorem and some manipulation will produce a derived system with at least N solutions and one less term. We are done, since when N = this is impossible. QED By employing the quadrature options embodied in Theorem, we have been able to calculate various B n s), n s) values to extreme precision. One may also use the quadrature prescription to create contiguous plots of box integrals, as in Figure. Referring to Figure, note that both plots show the natural pole at s =, where the standard expectation integral diverges for n = dimensions. However, we see that also has poles at s = 3, as in Corollary. 3 Quadrature experiments We have employed the quadrature schema of Theorem to effect extreme-precision values. We might mention here that in fact this research began in earnest with the discovery, using the PSLQ integer relation algorithm see [5] and [8]), that the extreme-precision numerical value we were able to compute for 3 ), namely 3 ) = ) was experimentally) given by the formula? 3 ) = π 6 log + log + ) + log + ) 3 log + ) 3. 7) As in previous works, by extreme precision we mean, loosely speaking, enough precision to resolve a given entity into fundamental or at least previously studied) constants; in the modern era the age of, say, LLL and PSLQ methods this usually means precision to hundreds to thousands of decimal digits. 5

6 Soon we were able to prove this evaluation. 5 Upon further experimentation and analytic discoveries, we realized that not only do some B n for various n have constants in common such as log-surds, π, and so on but the n constants also have many constants in common with themselves and the B n. Such experimentation/detection moved us to find algebraic relations between the various entities, as we describe in this paper. Our numerical approach was to implement, using the Fortran-9 interface to the ARPREC arbitrary precision software package [], the four formulae of Theorem, namely ), 3), ) and 5), and the definitions of the functions bu), du) and Dρ, K) as given by formulae ), 3) and ). As stated in Theorem, these formulae are valid for all s except for positive even integers. For positive even arguments the integration in, ) is trivial and the relevant box integral is rational, obtainable quickly with symbolic processing. 6 Computation of bu) is entirely straightforward, once one has in hand an implementation of the erf function see [, pg ]). Computation of du) is complicated by the fact that for small arguments, cancellation of terms in the numerator typically results in severe relative numerical error. This can be ameliorated by employing the Taylor series for du): du) = u 6 + u 3 u u8 8 u 79 + u 655 8) ) n u n = n + ) n + )! n= Computation of the A k coefficients was achieved by employing an arbitrary precision version of the well-known Linpack program for solving linear equations via LU decomposition []. The four formulae ), 3), ) and 5) of Theorem themselves involve significant numerical difficulties, as they involve near-cancellation of terms for small arguments. Nonetheless we were able to compute the B n s) and D n s) to approximately 7-digit accuracy, utilizing 5-digit working precision, by using Gaussian quadrature, after splitting the integral into the intervals, ) and, ), and applying the simple substitution w = /u on the second integral. We should add here that tanh-sinh quadrature [], which we have used in numerous other studies of this sort, was not needed in this case, because each of the integrand functions in Theorem are regular on the intervals of integration. In addition, the tanh- 5 That is to say, the? = became =. In the present work, we use = to mean rigorously proven; indeed, some of our more recondite closed forms were first found in the? = sense, then later proven. 6 Mathematica and Maple were both used quite heavily for much of this paper s development, and most of the work was symbolic. We have endeavored to check all results numerically, even after obtaining exact, symbolic forms. That being said, elsewhere in this paper we have for efficiency eschewed comments about which computer algebra system was used where. 6

7 sinh scheme suffers here from the cancellation difficulties mentioned above, since it relies on evaluating the function very close to endpoints of the interval. The numerical values we obtained in this fashion were used in conjunction with the PSLQ integer relation algorithm to find many of the analytical evaluations presented in this paper. All results are summarized in the Appendix. As explained at the beginning of the Appendix, -digit numerical values of these constants are available on a website. Validation of expressions To confirm the accuracy of the results in the Appendix, the B and integrals were separately computed using 5-digit working precision, programmed with the C++ interface to the ARPREC package. Integrals were evaluated by formulae ), 3), ) and 5), using Gaussian quadrature, and imported into Mathematica as numerical values. A parser for LaTeX source provided with Mathematica 7. was used to import the analytical formulae from the Appendix in a machine-readable format. Conversions were made to use Mathematica built-in routines for computing the Lewin arctan integral, Clausen function and hypergeometric function. The accuracy of these formulae was then checked by subtracting the numerical values computed by quadrature with numerical values produced by Mathematica for the analytical formulae. In our first application of this method, four errors were found among the B n s) formulae and three among the n s) given in the Appendix. Fortunately, in all but two cases we were able to find the correct formula merely by running PSLQ on the vector consisting of B n s) or n s) and the component terms in the flawed identity. These results were then double-checked using Mathematica. Diagnosing the errors in the remaining two formulae, assisted also by Maple 3, led to significant enhancements in the final version of this paper. We observe that this type of checking computing extreme-precision numerical values for specific instances of left-hand side and right-hand side expressions in a formula, then comparing the results is very generally applicable and highly effective in disclosing bugs in papers presenting new mathematical identities. Its general adoption would prevent many mistakes from making their way into published papers. 5 Analytic continuation for the B n In our previous work [5] we demonstrated that the B n expectations can always be reduced by at least one dimension; for example, via vector-field algebra, B n can be obtained from an integral over the n )-dimensional unit cube, like so for n > : B n s) = n n + s r [,] n + r ) s/ D r. 9) 7

8 Note that already with this reduction we see that B n has a single, simple pole at s = n, as the integral exists for all complex s. With the advantage of hindsight, we define right now a general and quite useful function C n,k s, a) := r r r k a + r ) s/ D r, ) r [,] n and we also define C, s, a) := a s/. In this way we can begin developing relations for the B n, as follows: B s) = B s) = B 3 s) = s + C,s, ) = + s C,s, ) = s C,s, ) = s +, ) + s F, s ; 3 ) ;, ) s) + s) π/ + sec t) s/+ ). 3) Note that the apparent pole at s = for B 3 s) is specious; the /s + ) factor cancels in this case, to yield B 3 ) = 3 π/ log + sec t) dt. ) For n = and higher dimensions, such relations for the B n rapidly become unwieldy. Take n =, and note that C 3 s, ) is a 3-dimensional integral which yields, after integrating first in polar coordinates r, φ) to handle two of the dimensions, B s) = = + s C 3,s, ) 5) 8 π/ dφ + sec φ + z ) s/+ + z ) s/+) dz. + s) + s) Again the specious pole factor for s = cancels, giving the special case B ) = π/ dφ log + sec φ + z ) log + z ) ) dz. 6) There is another form of analytic relation also derived previously [5], namely B n s) = n+s/ n + s k s/) k β n,k, 7) n) k 8

9 where the β coefficients arise from the b function, implicitly as bu)e u) n =: β n,k k u k. 8) and a) k = Γa + k)/γa) is the Pochhammer symbol. The β numbers enjoy the relations here, d!! means 3 d for odd-positive integer d): β n,k = β n,k = k +...k n=k k j= k k + )!! k n + )!!, k + )!! β n,k j, + k/n) β n,k = β n,k + β n,k, 9) with the recursions here ignited by β,k := δ,k and β n, = n/3. The main point is, for n > the sum 7) is linearly absolutely convergent for all complex s. The summand decay factor is essentially /n) k.) Accordingly, the series gives the analytic continuation over the entire s-plane. This series representation can be used computationally, either in place of the quadrature formulae, or as an extremeprecision check on same. The analytic series does have one advantage over the quadrature formalism: One need not break the plane by polarity of Rs); the series 7) always converges and there is the correct single-pole factor /n + s)). 6 Theory of hyperclosure We have seen that the box integrals B n are expressible in terms of C-functions. It will turn out that there exist recurrence relations between C-functions; this leads eventually to relations between n and B n values, and closed forms in many cases. To quantify for our purposes what is a closed form, we first establish 7 Definition [Ring of hyperclosure] We define a hypergeometric evaluation in a classical sense to be a complex number n a nz n where z is complex algebraic, a is a real rational, and generally a n+ = rn) a n where r is a fixed rational-polynomial function ratio of two polynomials each with integer coefficients). We then contemplate the ring of hyperclosure, generated by all hypergeometric evaluations under ordinary operators, +). An element of this ring is said to be hypergeometrically closed hyperclosed for short). We also deem to be hyperclosed so that hyperclosure not be denied at poles). 7 It has occurred to the present authors that the age-old notion of closed form might be addressable, as a separate research program, vis such as our Definition. However, in no sense can Definition be all-encompassing; e.g., most would say that something like /π + e) is a closed form, yet for all we know it might not lie in our hyperclosure ring. 9

10 As examples, any rational a sum with one term, say r := ) is hyperclosed, and since = F,, ; ), π = F,, 3/, ), log + z) = z F,, ; z), it follows that combinations such as + π, π, π log 7, 3 log + ) are all hyperclosed. Relevant to the present work is the fact of the Lerch transcendent Φz, s, a) := n z n n + a) s, 3) being hyperclosed for algebraic z, integer s, and rational a. Thus, we shall have hyperclosure for any polylogarithm Li s z) := n zn /n s for integer s and algebraic z, so perforce for the Lewin arctan integral Ti z) := ) n z n+ = z n + ) Φ z,, ), 3) n and the Clausen function Cl z) := n sinnz) n. 3) Our box-integral tables have hyperclosed entries, except where dangling integrals remain when dimension n >. Although we admit such unresolved integrals may still belong to the ring.) An interesting sidelight looms here: What numbers are not hyperclosed? Certainly we need such numbers to exist, lest our entire research program of finding hyperclosed expressions be a vacuous exercise. Given the algebraic character of parameter z in Definition, and the constraint that the succession ratio rn) be rational-polynomial, together with the fact of all group generations under, +) yielding finite strings, we conclude that the ring of hyperclosure is countably infinite. Therefore: Almost all complex numbers are not hyperclosed. Equivalently, the ring of hyperclosure is a null set in the complex plane. As often happens in such analyses, we are stultified by the question: If non-hyperclosed numbers are so overwhelmingly abundant, what is an example of such a number? Well, we do not presently know a single such number. Perhaps π π is not hyperclosed, but such questions loom Hilbertian in their evident profundity. 8 It is evident that the lowest-lying C-functions allow direct hypergeometric evaluation, so that B s), B s) are hyperclosed for any integer s, but that already for C, we run into some difficulty with the integration. Happily, there are some powerful connection formulae that we prove presently, starting with 8 Such a radical disconnect between theory and knowledge occurs elsewhere; e.g., in the study of normal numbers. Though almost all real numbers are absolutely) normal, only artificially constructed normals are known.

11 Theorem [Convergent series for C-functions] For all complex s, integer n >, and Ra) > we have C n, s, a) = n + a) s/ k ) k s/) k β n,k, 33) n + a which expansion being linearly convergent with k-th summands being O n )/n + a)) k). Proof: This expansion can be established using the same methods as for expansion 7), which is the case a = as shown in [5]. QED And now for some key recurrences: Theorem 3 [Fundamental relations for C-functions] Special instances of the C- function are C n, s, ) = B n s), 3) C, s, a) := a s/, 35) C n, s, ) = n + s + n + B n+s). 36) More generally, for all complex s, positive integer n, and positive real a we have asc n, s, a) = s + n)c n, s, a) nc n, s, a + ). 37) Another recurrence, involving now the second index on C, is, for positive integers a, k, sc n,k s, a) = C n,k s, a + ) C n,k s, a). 38) Proof: The recurrence 37) is proved by invoking β-relation 9) to show the right-hand side here in the theorem is a C n, s, a)/ a) which equals the left-hand side. The recurrence 38) is easier, requiring simple integration by parts in the integral representation ) for C. QED We are now prepared to establish a central result that will lead to a host of closed forms, which result being: Theorem For any integer s and positive integer a, all the numbers C n, s, a) := a + r ) s/ D r 39) r [,] n for n =,, are hyperclosed.

12 Remark: As with other parts of the present treatment, we not only prove hyperclosure, but do this constructively, so that an algorithm for achieving closed forms is implicit in the proof. Proof: C, s, a) := a s/ and C, s, a) = a s/ F, s/; 3/, /a) are clearly hyperclosed. So we turn our attention to C, : For nonnegative even integer s, it is evident that the defining integral for C, yields rational values. Consider, then, a single evaluation for negative even s, where we employ polar coordinates in the defining -dimensional integral to obtain where π/ C,, a) = dt loga + sec t) log a ) ) ) + a + a = G + Ti + π a log + ) + a π log a, G := n ) n n + ) = Ti ) ) is the Catalan constant and as before Ti is the Lewin inverse-tangent integral; see 3, 3). Clearly C,, a) is hyperclosed for positive algebraic a. Now, in recurrence 37) with n =, there is a term C, s, a + ) which we have seen to be hyperclosed. The recurrence thus reveals that C, s, a) is hyperclosed for all negative even integers s. It remains to handle all odd integers s. We observe another ignition value ) C,, a) = ) 3 a3/ tan + ) a + a a + a 6 3a + ) log + a +, a + 3 and this again drives the recurrence 37) to yield hyperclosure for any odd s. QED 7 Hyperclosure and the B n Theorem 5 [Interrelations for the B n ] For integer n > and all complex s, we have a recurrence n + s)n + s )B n s) = sn + s )B n s ) + nn )C n, s, ), 3) as well as various higher-order recurrences enjoying reduced indices on the C-functions, for example s )ss + ) B 5 s ) s + 3) {s3s + 7) B 5 s ) s + )s + 5) B 5 s)} ) = 6C, s, 3).

13 In addition, the residue Res n of the unique pole at B n n) is given for positive integers n by Res n = nc n, n, ) 5) = n B nn ) n n + ) + n C n, n, ) ; n >. 6) Finally, we have a special analytic-continuation value valid for all positive integers n: B n n ) = nc n, n, ). 7) Remark: In the instance s = n where a pole is involved, an expression n + s)b n s), on the face of it, is to be interpreted as the residue Res n at the relevant pole. These residues allow us to effectively traverse B n -poles when applying recurrence 3). Proof: The first B n, C recurrence 3) follows from the combination of proven relations 36) and 37). The second, higher-order recurrence is obtained by combining 37, 3). The residue formulae follow from 3) and 36). QED Theorem 5 allows us to evaluate important residues Res n, at least for low-lying n. In fact, we remind ourselves that Res n := lim ǫ B n n+ǫ), and state the residue evaluations that were essential in rigorously establishing our tables of closed forms: Res =, Res = π, Res 3 Res 5 = π, Res = π 8, = π. Proof: The evaluations for n =, are easy, using the first residue relation in Theorem 5. We then have Res 3 = B 3 ) + 6C,, ), Res = B ) + 6C,, ), Res 5 = 3 B 5 3) B 5 ) + C,, 3). Each of these boils down to a rational times a power of π we note that all four B values here are independently resolvable). QED Incidentally, we believe that Res 6 = π 3 /6, and, remarkably, O-Y. Chan has recently conjectured a result that implies Res 7 = π 3 / with similar results for Res 9,,.... In 3

14 any case it seems reasonable to conjecture that in general, Res n = Q n π n/, where Q n is rational. Note that a proof of such a general conjecture would essentially yield an evaluation of any C m, m, ), for example, the specific conjecture about Res 7 guesses a closed-form evaluation C 6, 7, ) :=? = π3 8. ) + x +...x 7/ 6 dx dx 6 We are now in a position to state a summary theorem about the box integrals B n. Theorem 6 [Hyperclosure for specific B n ] For any integer s, each of: B s), B s), B 3 s), B s) is hyperclosed. As for 5 dimensions and integer s, B 5 s) is hyperclosed except perhaps for s =,. Remark: Again the proof following is constructive; i.e., gives rise to an immediate algorithm for generating closed forms. Note the exceptions s =, for n = 5 dimensions; these exceptions represent the boundary of our current understanding of box integrals, even though these difficult cases may well be hyperclosed themselves. Proof: From Theorem it is immediate that B m s) = m C m+s m,s, ) is hyperclosed for m =,, 3. So we turn to B, and observe that for nonnegative even integer s, the value B s) is rational. From the first recurrence in Theorem 5, and knowledge of Res, we may take one value of B odd) and propagate that to all other B odd). The same goes for even arguments. Therefore, if we establish hyperclosure for B ) and B ), say, then we know all B integer) to be hyperclosed via Theorem. To these ends, we observe B ) = log 3 3 G + Ti 3 ) ) 8 arctan 8, 8) and B ) = π log + ) 3 G π 8. 9) A proof of 9) starts with 5) and the evaluation of the jellium value J given in our previous paper [5]; said proof thus reduces to showing log 3 + s ) ds = π + s log + ) 3 + π log Ti + ) 3 G.

15 Using A.3) in [9] this becomes log 3 + s ) ds + π + s 6 log + ) 3 π log 3 G =. Now substituting s = tan θ yields an expression which computer algebra evaluates as 3 G + π 3 log + ) 3 Ti + ) 3 Ti ) 3 =. This is easily confirmed using [9, A.3),)]. The proof of 8) is similarly completed; said proof reduces to resolving 7). Finally we handle the difficult case n = 5. One can prove that B 5 ) = π ) ) 3arctan log + 7 { K π 8 log + )} 3, and B 5 ) = 5 3 { K π log + )} π 5 ) 6arctan + 5 log 5 ) 5 +. where K is a certain definite integral that has been resolved; see our later section on K-integrals. Now, employing recurrence ), together with the known residue Res 5, we can propagate this knowledge of B 5 ±) to all B 5 odd). As for even integer arguments, note from our tables that B 5 6), B 5 8) are independently resolved in terms of a hyperclosed integral K. But, via the second recursion in Theorem 5, this means every B 5 k) is hyperclosed for even integer k 6. Thus the only exceptional instances out of all B 5 integer) are B 5 ), B 5 ) as claimed. QED Another approach to recursions for B n, n as a function of s for all positive integer n) is via the linear differential equations satisfied by b and d; see [] and [, pg. 7]. This leads, for example, to an implementable 5-term recursion for B s) in terms of B s ), B s ), B s 6), B s 8) with coefficients polynomial in s. 8 Hyperclosure and the n On the face of it, our definition ) of the box integral n would appear to present a n)-dimensional integration problem. But a certain n-fold integral suffices; in fact we have n n s) = n r k ) D r. 5) r [,] n r s 5 k=

16 This reduction from n to n dimensions can be derived in various ways. Perhaps the most intuitive is to look at the probability density of a coordinate difference x = r k q k, which turns out to be a triangle distribution with base [, ] and height at the origin-centered apex; i.e., the density is x, and the new integral form follows. It is interesting that 5) has leading term n B n s) arising from the component of the integrand s product, indicating heuristically that n and B n should somehow be analytic relatives. Indeed, one can carry such an idea much further, to end up with powerful analytic expressions for n s). Simple symmetries of the integrand in 5) reveal that n ) n n s) = n ) k C n,k s, ). 5) k k= On the other hand, we have the recursion 38) that may be used to decrement the both indices n, k on C n,k. Thus we have the intriguing principle that n can be formally expressed in terms of C m, functions, where m n. But there is more: Some thought in regard to relations 36, 37, 3) reveals that the relevant C m, values lead back to the B n box integrals, so that n can be formally expressed in terms of the B n. The summary result is: Theorem 7 [ n as an analytic superposition of B,...,B n ] For arbitrary positive integer n and any complex s, we have an analytic superposition n m n s) = E n s) + R m,j s)b m s + n j), 5) m= j= where E n is an elementary function and each R m,j is a rational-polynomial function. Remark: Thus n is in general a superposition of an E term and n + )n )/ different B terms. Proof: This result follows from the combinatorics embodied in relations 5, 36, 37, 3). QED Let us now give some entirely general analytic n evaluations; first the easy case n =, for which the sum in Theorem 7 is empty and we have a simple, elementary function: s) = x)x dx = + s + s. 53) Note the two poles, as we expect from previous analysis. Then from the combinatorics leading up to Theorem 7 we obtain 3 + s) s + + s) = 8 s + )s + 3)s + ) + B s) 6 s + ) s + B s + ). 5)

17 Note now the existence of three poles at s =, 3, ). Going yet deeper into the combinatorics, we have s 3 + s + s +3) s + 5) + 3 s) = s + )s + )s + 5)s + 6) + s + B s + ) s + 6) s + )s + ) B s + 5) s + ) + 8B 3 s) s + B 3s + ) s + 6)s + 7) + s + )s + ) B 3s + ). 55) There are some difficulties attendant on this 3 form; namely, the apparent pole at s = is specious. In order to get such as the 3 ) closed form in our tables, one needs to take a careful limit s. Note also that 3 pole at s = 3 is caused exclusively by the 8B 3 s) term itself. Also, one needs a residue evaluation from Theorem 5 for the B 3 s+) term at s = 7. At the next level we obtain the complete analytic continuation for, in terms of B, B 3, B,: s s+6 3 s +) s + 7) + s) = 6 s + )s + )s + 6)s + 7)s + 8) 96 + s + )s + ) B 96s + 8) s + ) s + )s + )s + 6) B s + 6) + 6 s + B 3s + ) 96s + 7) s + )s + ) B 3s + 8)s + 9) 3s + ) + s + )s + )s + 6) B 3s + 6) + 6B s) 88s + 6) 3s + ) B 8s + 8)6s + 3) s + ) + 3s + )s + ) B s + ) 8s + 8)s + 9)s + ) 3s + )s + )s + 6) B s + 6). 56) Again the apparent pole at s = is specious, so one must work out another logarithmic limit in that case. For s = 9, we may use the known residues of the B-poles. The poles at s =, 5, 6, 7, 8 do exist; e.g. the term with B 3 s + ) has a pole at s = 5.) Finally, as with lower dimensions, we can supply the following superposition formula for 5 in terms of B n with n 5, 7

18 5 s) = s)6+s/ + +s 5 +s/ 3 5+s/ ) 57) + s) + s)6 + s)8 + s)9 + s) + s) s) + s)6 + s) B 3 + s) 6 + s) + s) + s)6 + s)8 + s) B 8 + s) s) + s) B s) 3 + s) + s) + s)6 + s) B s) 6 + s) + s) + + s) + s)6 + s)8 + s) B s) s B + s) s) 3 + s) + s) B + s) s)55 + 6s) 3 + s) + s)6 + s) B 6 + s) 8 + s) + s) + s) 3 + s) + s)6 + s)8 + s) B 8 + s) 7 + s) + 3B 5 s) 6 + 3s B 5 + s) s) s) B 5 + s) 3 + s) + s) s) + s)7 + 5s) B s) + + s) + s)6 + s) 3 + s) + s) + s)3 + s) B s). + s) + s)6 + s)8 + s) Such analytic formulae allow us to state a companion theorem to Theorem 6, as: Theorem 8 [Hyperclosure for specific n ] For any integer s, each of: s), s), 3 s), s) is hyperclosed. As for 5 dimensions and integer s, 5 s) is hyperclosed except perhaps for s =,,. Remark: A summary of results here: With Theorem 6, we now know that B,,3, and,,3, are all hyperclosed at every integer argument, and that all B 5 integer), D 5 integer) are now proven hyperclosed, except for the short exception list B 5 ), B 5 ), 5 ), 5 ), 5 ). It is especially interesting that the resolution of just one of B 5 ), B 5 ) would settle this whole exception list, because a certain linear combination of these two B 5 s is hyperclosed. Proof: All of these hyperclosure results follow from the above analytic superpositions for n in terms of B s, together with knowledge of all relevant residues Res n, n =,, 3,, 5. In particular for n = 5, the terms in 57) involving B 5 s have arguments s, s+, s+, s+ 6, s+8, so that whenever integer s {,, }, only resolved B 5 s are involved. The established the set of three unresolved 5 s. QED 8

19 9 Difficulties in n 5 dimensions We have not been able to obtain hyperclosure for all integer arguments for n = 5 because of the finite exception list see the Remark after Theorem 8). There may be some future hope in employing π/ π/ + s) + s)b 5 s) = sec a) + sec b) + z ) +s/ dz dadb π/ π F, s ; 3 ) ; cos b) sec +s b) db + π 5 + s. 58) For integer s the integral in z can be performed symbolically and the hypergeometric function evaluated. This reduces B 5 s) to a double integral expression for s = 6; and also for s =, on applying the l Hôpital rule. As for the sultifying scenario of n = 6 dimensions, we admit right off that we know not a single nontrivial closed form for a B 6. One might conceivably use B 6 s) = π 8 s + )s + ) s + 6) 3 s + B s + ) π valid for all values of s. In particular π/ π/ π/ 3 s + )s + ) B s + ), π/ π/ π/ B 6 7) = ) 5 arctan 8 Moreover, the triple integral above resolves to π/ π/ + π 8 { arctan sec a + sec b + sec c ) 3+s/ da dbdc da dbdc sec a) + sec b) + sec c) π π. arcsec + sec a) + sec b)) da db = 3 arcsec z + 3)arcsec z) sec a) + sec b) z + ) 3/ z + ) ) arctan + 8 arctan 6) }. Thus we arrive at B 6 7) = 5 K π 7 ) 5 π arctan + 8 ) 8 5 π arctan 3 ) 5 π arctan 59) where an integral form of K 8 an integral still unresolved, to our knowledge is exhibited in the next section. dz 9

20 K-integrals In a set of marvelous preprints, G. Lamb [7] managed to resolve some dangling integrals from our previous work [5]. We called these K m, for m =,,, 3. Now, via the combination of proof via alternative integration as with result 7) below), together with the powerful Lamb techniques, we know closed forms such as K := arctanh 3+y ) + y dy = 3 Ti 3 ) + π log + ) G. 6) Thus K is hyperclosed. The next integral is perhaps the most difficult one of the present treatment. Along the way to the following answer having started with the Lamb method were stages of symbolics, at times for us involving expressions of over characters in modern symbolic processing. where arcsec x) K := 3 x x + 3 dx = Cl θ) Cl θ + π ) Cl θ π ) + Cl θ π ) Cl 3 θ + π ) Cl 3 θ + π ) Cl 3 θ 5π ) + Cl 3 θ + 5π ) θ 5 π ) log ) 3. 6) 3 θ := arctan 6 3 ) 5 + π. It may well be that this closed form for K can be further simplified. Not so hard, but certainly nontrivial, is ) π/ K := + sec a) arctan da 6) + sec a) = Ti + ) 3 + π 8 log + ) 3 + π 3. Finally, we have resolved a dangling integral from [5] involving dimension n = 5, as K 3 := π/ π/ + sec a) + sec b) dadb = log + 3) + 8 π B 5 3). 63)

21 Indeed the formula for the jellium constant J 5 given in [5, 7.3] yields precisely 63) and the closed form for B 5 3) is listed in Table. Moving on, we hereby define additional K-integrals, starting with ) π/ arctan +sec K := t) dt 6) + sec t) and K 5 := π/ π/ log + sec a) + sec b) ) dadb. 65) Following suggestions in [8] and significant symbolic computation, we are express K explicitly in terms of complex dilogarithms. The computer algebra system produced an answer, which with considerable massaging became 8 K = 7π 6 3π arctan ) + 6 arctan 7 + ) 3π 6 )} + 8 R { Li + i ) + { R Li i R { Li 5 5 i 5 Li + i 66) ) 5 i Li i + 3 )} i ) 5 i Li 5 5 i + )} i. The arguments of the dilogarithms here solve an interesting, degree- polynomial system. One may use dilogarithm reflection formulae to obtain closed forms such as ) ) ) ) K = Li 3, θ + Li, π θ Li, π α + Li 3, β ) + π 8θ) arctan, 67) where ) 3 + ) θ := arctan, α := arctan + 3 and 3 + ) β := arctan + 3, with the Lewin generalization Li r, φ) := RLi re iφ ) ; see [9, A.5 )]. It is immediate that K is hyperclosed, since in our K formulae any arguments re ±iθ are all algebraic, and RLi z) = Li z) + Li z ).

22 So, we know that K,,,3, are all hyperclosed but we do not know this for K 5. Various recondite, but alas partial results can be derived, starting typically from 65) and employing coordinates tana, tanb over the unit -square. One can obtain such as a - dimensional-integral dependence arctan t ) ) arctan 3+t K 5 = dt π G + π log 3 68) + π or an efficient sum K t + t) 3 ) 8 Ti 3 ) 8 + π Ti = π 6 log 3 m3 m m m j= m j 3 ) h j π 6 ) + π 8 log ), ) h m j π ), where h k := /3+/5 ±/j ), with h :=. This summatory representation allows extreme-precision evaluation of K 5 without recourse to quadrature per se. Now to the concept of alternative integration: It is highly interesting that the boxintegral theory can sometimes be used to resolve previously unknown integrals that have, on the face of it, little to do with n-dimensional boxes. A canonical example is the case of 3 ), which is obtained through the rather delicate limit process s in 55) in order to complete hyperclosure for n = 3. But when we attempted to evaluate 3 ) from the alternative formula 3 ) = 8 r [,] 3 x) y) z) x + y + z dx dy dz, 69) we ended up with a single, troublesome arctan integral as the dangling term. But the s ) limit procedure does a complete hypergeometric breakdown, and all of this proves the result ) K 6 := + z arctan dz 7) + z = G + π log + ) + Ti 3 ). We do not give here a direct proof of this integral relation, although G. Lamb has now provided a fine analytic evaluation of K 6 ; see [8]. Another example of an indirect integral resolution is π/ K 7 := log + + sec t) dt = G + Ti 3 ) + π log + ). 7)

23 Finally, we define K 8 := arcsec x + 3)arcsec x) dx, 7) x + ) x + ) 3 and observe that K 8, along with the elusive K 5, are the only unresolved instances of our integrals K,,...,8. Again, knowledge of K 8 would give us B 6 7), as in 59), while hyperclosure of K 5 would establish hyperclosure of all B 5 integer), 5 integer). Curiosities One new result of some interest is that another unphysical, but valid analytically continued value discussed in our previous work [5], namely B 5) = is now known in closed form. In fact, this previously mysterious analytic-continuation value is 8arctan/ 8) see Tables and the last part of Theorem 5). However, there are still many unexplained empirical phenomena. For example we do not know in closed form the zero of s) at s = , though we have developed herein enough machinery to find this zero to extreme precision. See Figure for a pictorial of the zero.) Moreover, we are not aware of any other zeros in n = or any other dimension; evidently, this zero of is some kind of anomaly. There is also an empirical local minimum for B, at s = , and again a closed form is unknown. Another curiosity is that some statistical quantities over the hypercube are almost trivial to resolve. For example, the exponential expectations E n λ) := e λ r r [,] n, F n λ) := e λ r q r, q [,] n, for constant λ do not need to be obtained from a generating function or series or anything complicated. Instead, closed forms are immediate, based on the defining relations 3, ): π ) n E n λ) = b n λ) = erf n λ), λ 3

24 F n λ) = d n λ). The allure of such closed forms is that they hold in all dimensions, and may well replace the box integrals B n s), n s) as the right tools to assess the statistical character of point clouds, again as in reference []. In spite of this streamlined approach to cloud measures, it seems heuristically) that both of e κ r r [,] n, e κ r q r, q [,] n should be extremely difficult to evaluate in any general way. Even if one were to revert to series expansion of the exponential, one would need B n s) for all nonnegative integers s, yet we have met with extreme difficulties when n > 5. This is not to claim some clever prescription a novel use of recurrences, or differential relations would not solve this problem. And what about noninteger arguments for the box integrals? One might extend the definition of hypergeometric closure based on such findings as B 3 7 ) = π 8 5 F 5 ;, 3 ; 9 ; ), Γ ) 9 π Γ 7 73) 5 ) F, 3 ; 7 ) ;. Here, F is the Appell hypergeometric function defined as a series in two variables []. Thus we expect entities such as B 3 odd/) to involve the Appell hypergeometric. Such examples suggest that there might be a kind of higher level of hypergeometric-closure theory for noninteger rational arguments s. Conjectures and open questions We conclude by reiterating the following open questions.. Is it a reasonable conjecture that every box integral B integer integer), integer integer) be hyperclosed? That is, could it be that our results for n actually are extensible for all positive integer n?. Is there a general evaluation hyperclosed or not) of B n n ) for all natural numbers n? 3. What transpires for noninteger but rational s in dimension n 5 see relation 73))?. Can K 5 be resolved? As we have said, this is the sole blockade to proof of hyperclosure for all B 5 integer), 5 integer).

25 5. Referring to Figure, what is a closed form for the mysterious zero? Are there any other possibly complex) zeros of any n whatsoever? Is there a closed form for the local minimum of B s) at s ? 6. What is an explicit number that is not hyperclosed? Even though almost all numbers are not hyperclosed, we presently do not know a single such number. We have suggested that perhaps π π is not hyperclosed; moreover, in the absence of known hypergeometric forms for the Euler constant γ, one is tempted a to conjecture that it, too, is not hyperclosed. Acknowledgments. The authors are grateful to T. Wieting for clarifications on hypergeometric algebra, as well as to O-Y. Chan, G. Lamb, J. Philip, M. Trott, and E. Weisstein for their contributions to box-integral theory. In particular, the Clausen terms in earlier formulae for ), 5 ) in [5] needed repair; we thank J. Philip for pointing this out in his delightful paper []. 9 Finally, the authors are deeply indebted to A. Kaiser of U.C. Berkeley and LBNL for writing computer code to check the formulae presented in the Appendix. References [] R. Anderssen, R. Brent, D. Daley, and P. Moran, Concerning x + x n) dx dx n and a Taylor series method, SIAM J. Applied Math., 3 976), 3. [] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, NBS now NIST), 965. See also [3] David H. Bailey and Jonathan M. Borwein, Highly parallel, high-precision numerical integration, Int. Journal of Computational Science and Engineering, accepted Jan. 8. Available at [] David H. Bailey and Jonathan M. Borwein, Effective error bounds for Euler-Maclaurin-based quadrature schemes, th Annual HPCS Proceedings, IEEE CD, May 6. Available at [5] D.H. Bailey, J.M. Borwein and R.E. Crandall, Box integrals, Journal of Computational and Applied Mathematics, 6 7), [D-drive Preprint 3]. [6] D.H. Bailey, J.M. Borwein and R.E. Crandall, Integrals of the Ising class, J. Phys. A., 39 6), 7 3. Available at 9 Therein his formula for ) has yet a newer misprint. As mentioned in Section, the present authors believe that the best way to ascertain what version of a recondite closed form be the correct one across publications, say is to test expressions via extreme numerical precision. 5

26 [7] David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor, and Eric W. Weisstein, Ten problems in experimental mathematics, Amer. Mathematical Monthly, 3 6), Available at [8] David H. Bailey and David J. Broadhurst, Parallel integer relation detection: Techniques and applications, Mathematics of Computation, 7 Oct ), [9] David H. Bailey, Xiaoye S. Li and Karthik Jeyabalan, A comparison of three high-precision quadrature schemes, Experimental Mathematics, 5), Available at [] David H. Bailey, Yozo Hida, Xiaoye S. Li and Brandon Thompson, ARPREC: An arbitrary precision computation package, Sept. Available at [] Jonathan M. Borwein and David H. Bailey, Mathematics by Experiment, AK Peters, 3. Second edition, 8. See also [] J. M. Borwein and B. Salvy, A proof of a recursion for Bessel moments, Exp. Mathematics, vol. 7 8), 3 3. [3] Chaunming Zong, The Cube-A Window to Convex and Discrete Geometry, Cambridge Tracts in Mathematics, 6. [] R.E. Crandall, C. Joe, and T. Mehoke, On the fractal distribution of brain synapses, manuscript, 9. [5] Helaman R. P. Ferguson, David H. Bailey and Stephen Arno, Analysis of PSLQ, an integer relation finding algorithm, Mathematics of Computation, 68 Jan 999), [6] Wolfram Koepf, Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities, American Mathematical Society, Providence, RI, 998. [7] G. Lamb, An evaluation of the integral K, A method of evaluating the integral K, and An approach to evaluating the integral K 3, preprint collection, 6. [8] G. Lamb, An evaluation of the integral K 6, and An approach to evaluating the integral K, preprint collection, 9. [9] Leonard Lewin, Polylogarithms and Associated Functions, North Holland, 98. [] Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart, LINPACK User s Guide, SIAM, Philadelphia, 979. [] J. Philip, The distance between two random points in a - and 5-cube, preprint, 8. [] D. Robbins, Average distance between two points in a box, Amer. Mathematical Monthly, ), 78. 6

27 [3] Charles Schwartz, Numerical integration of analytic functions, Journal of Computational Physics, 969), 9 9. [] Hidetosi Takahasi and Masatake Mori, Quadrature formulas obtained by variable transformation, Numerische Mathematik, 973), 6 9. [5] M. Trott, Private communication, 5. [6] M. Trott, The area of a random triangle, Mathematica Journal, 7 998), [7] E. Weisstein, Hypercube line picking. Available at 7

28 3 Appendix: Compendium of proven closed forms Numerical tables in support of the following closed forms are to be found at 8

29 n s B n s) any even rational: B ) = /3 - any s+ - π log + ) 3 + ) ) any +s F, s ; 3 ; ) Table : Some relatively easy evaluations for B n s). Every B k integer) for k =, can be given a closed form see Theorem 6). 9

30 n s B n s) π arctan 3-3G + 3 π log + ) + 3 Ti 3 ) 3 - π + 3 log + 3 ) ) ) 6 3 any Integral representation 3), relation 36) Table : Example evaluations of B 3 s). Every B 3 integer) can be given a closed form, being as we can express every B 3 in terms of a function C, which we have shown to be hyperclosed at relevant arguments. 3

31 n s B n s) -5 8 arctan - ) 8-3 G Ti 3 ) - π log + 3 ) G π 8 - log 3 G + Ti ) arctan G + 3 Ti ) log 3 7 arctan any Integral representation 5), recursions 3) ) 8 ) 8 Table 3: Example evaluations of B s). Every B integer) can be given a closed form, on the basis of recursions and ignition values such as the pair B ), B ). 3

32 n s B n s) K 36 5 π ) arctan 88 5 arctan ) 7 ) arctan K 5 π arctan π ) 5-5 K 5 5 π G + 5 π log + ) + 5 π Ti ) G log 3 ) θ 9 8 π Cl θ + π) + Cl 3 3 θ π) Cl 3 θ + π) + Cl 6 3 θ + π) Cl 3 3 θ + 5 π) Cl 3 3 θ + π) B 3 5 6) B 3 5 ) + 5 π log 3 + Ti 3) G 5 - G + θ log 3 ) π log ) 5 5 3arctan + Cl 3 θ + π) Cl θ π) 3 6 Cl 9 θ + π) Cl 6 9 θ + π) + Cl 3 9 θ + 5 π) + Cl G + 7 θ log 3 ) π log Cl 9 θ + π) 7 Cl Cl 7 θ + π) Cl 6 7 θ + 3 π) Cl + 5 ) 3 θ π) 3 6 θ + 5 π) + Cl 3 7 ) 5 θ + 6 π) 3arctan θ + 6 π) 5 any odd Closed forms follow from recursion 3), using residues as needed 5 any even Closed forms follow from recursion 3), using residues as needed, with only B 5 ), B 5 ) remaining unresolved. ) 5 Table : Example evaluations of B 5 s). Here θ = arctan6 3 5)/). We have proven that every B 5 odd) is hyperclosed. Note that K here is known, hyperclosed; yet, an unresolved dangling ) integral K 5 is not yet known to be hyperclosed, but may well be. A hyperclosure proof for K 5 would serve to establish all the B 5 even) by settling B 5 ), B 5 ), leading thus to resolution of all B n integer), n integer) for n =,, 3,, 5. 3

33 n s n s) any even rational: ) = /3 -, - any s+ s ,-3, log + ) ) 3 any Formula 5), using known B n -residues as needed Table 5: Some relatively easy evaluations for n s). Every k integer) for k =, can be given a closed form see Theorem 8). Note: The closed form for ) here has been repaired w.r.t. the previous work [5].) 33

34 n s n s) π 5 3 3,-,-5,-6 ) ) 3 π G + Ti 3 + 6π log + + log 5 log 3 8 arctan 3 - π + ) log + + log + ) 3 log + 3 ) 3 8 π + ) log log + ) π log + ) + 3 log + ) any Formula 55), using known B n -residues as needed ) Table 6: Example evaluations for 3 s). Every 3 integer) can be given a closed form see Theorem 8). 3

35 n s n s),..., π 8 log + ) 3 log + 3 ) + 6 log + log ) ) arctan 8 96 Ti G π log + 6 π log + 3 ) π + π + ) arctan + log 3 π log ) ) + 8Ti log ) log + 3 ) 8 G + 8Ti ) ) arctan / 3 6 π log + log log + ) + log + 3 ) ) arctan 8 G 5 + Ti ) 5 3 any Formula 56), using known B n -residues as needed 5 Table 7: Example evaluations for s). Every integer) can be given a closed form see Theorem 8). 35

36 n s n s) 5-5,..., G π π log 3 + log + ) + log + 3 ) + 6 log + ) ) ) arctan 8 8 3arctan 5 + log 3 ) θ Ti ) Cl 63 θ + π) Cl θ π) 3 6 Cl 89 θ + π) 8 Cl 6 89 θ + π) + Cl 3 89 θ + 5 π) + 8 Cl 3 89 θ + π) 6 5 any odd Closed forms follow from 57), using residues as needed 5 any even Closed forms follow from 57), using residues as needed, with only 5 ), 5 ), ) remaining unresolved. Table 8: Example evaluations of 5 s). It is proven that every 5 odd) is hyperclosed. Every 5 even) is hyperclosed, with perhaps three exceptions as in the table. 36

37 Figure : Analytic continuation: Plots of B s) top) and s) bottom) for real s in various intervals. B s) is the expected value of r s where r is distance from origin to a random point on the unit square [, ]. Likewise, s) is the expectation with r being the distance between two random points. B s) has a solitary pole at s =, while s) has three poles, at s =, 3,. Evidently, s) also has a mysterious zero at some real s see text). 37