Closed Forms: What They Are and Why We Care

Jonathan M. Borwein and Richard E. Crandall

Closed Forms: What They Are not be considered closed-form. However, abounds in terms that are in frequent the choice of what to call closed-form use yet are rarely made precise. Two such are rigor- and what not is rather arbitrary since a ous proof and closed form (absent the technical use new “closed-form” could simply within differential algebra). If a rigorous proof is be defined in terms of the infinite sum. “that which ‘convinces’ the appropriate audience,” —Eric Weisstein then a closed form is “that which looks ‘funda- There is not much to disagree with in this, but mental’ to the requisite consumer.” In both cases, it is far from rigorous. this is a community-varying and epoch-dependent notion. What was a compelling proof in 1810 may Second Approach. The next attempt follows a well not be now; what is a fine closed form in 2010 September 16, 1997, question to the long-operating may have been anathema a century ago. In this “Dr. Math” site1 and is a good model of what article we are intentionally informal as befits a interested students are likely to be told. topic that intrinsically has no one “right” answer. Subject: Closed form solutions Let us begin by sampling the Web for various Dear Dr. Math: What is the exact mathe- approaches to informal definitions of “closed matical definition of a closed form solution? form”. Is a solution in “closed form” simply if an expression relating all of the variables can Definitions be derived for a problem solution, as op- First Approach to a Definition of Closed Form. The posed to some higher-level problems where first comes from MathWorld [56] and so may well there is either no solution, or the problem be the first and last definition a student or other can only be solved incrementally or numeri- seeker-after-easy-truth finds. cally? An is said to be a closed-form Sincerely, …. solution if it solves a given problem in terms The answer followed on September 22: of functions and mathematical operations This is a very good question! This matter from a given generally accepted set. For has been debated by mathematicians for example, an infinite sum would generally some time, but without a good resolution. Jonathan M. Borwein is professor of mathematics at the Some formulas are agreed by all to be University of Newcastle. His email address is jonathan. “in closed form”. Those are the ones which [email protected]. contain only a finite number of symbols, Richard E. Crandall is professor of Mathematics at Reed and include only the operators +, −, ∗, /, College. His email address is [email protected]. DOI: http://dx.doi.org/10.1090/noti936 1Available at http://mathforum.org/dr/math/.

50 Notices of the AMS Volume 60, Number 1 and a small list of commonly occurring Tim Chow explains nicely why he eschews functions such as nth roots, exponentials, capturing all algebraic numbers in his definition, , , inverse why he wishes only elementary quantities to have trigonometric functions, greatest integer closed forms, whence he prefers E to Ritt’s 1948 functions, factorials, and the like. definition of elementary numbers as the smallest More controversial would be formulas algebraically closed subfield L of C that is closed that include infinite summations or prod- under exp and log. His reasons include that: ucts, or more exotic functions, such as Intuitively, “closed-form” implies “explicit”, the Riemann zeta function, functions ex- and most algebraic functions have no simple pressed as of other functions explicit expression. that cannot be performed symbolically, Assuming Shanuel’s conjecture that given n functions that are solutions of differential complex numbers z1, . . . , zn which are linearly in- (such as Bessel functions or hy- dependent over the rational numbers, the extension pergeometric functions), or some functions field defined recursively. Some functions whose Q (z , . . . , z , ez1 , . . . , ezn ) values are impossible to compute at some 1 n specific points would probably be agreed has transcendence degree of at least n over the not to be in closed form (example: f (x) = 0 rationals, then the algebraic members of E are if x is an algebraic number, but f (x) = 1 if exactly those solvable in radicals [31]. We may x is transcendental. For most numbers, we thence think of Chow’s class as the smallest do not know if they are transcendental or plausible class of closed forms. Only a mad version not). I hope this is what you wanted. of Markov would want to further circumscribe the class. No more formal but representative of many dictionary definitions is:

Third Approach. A coauthor of the current article In an increasingly computational world, an ex- is at least in part responsible for the following brief plicit/implicit dichotomy is occasionally useful, definition from a recent mathematics dictionary but not very frequently. Often we will prefer com- [17]: putationally the numerical implicit value of an algebraic number to its explicit tower of radicals, closed form n. an expression for a given and it seems increasingly perverse to distinguish function or quantity, especially an , the root of 2x5 − 10x + 5 from that of 2x4 − 10x + 5 in terms of known and well understood or to prefer arctan(π/7) to arctan(1). We illustrate quantities, such as the evaluation of these issues further in Examples 7, 9, and 14. Z ∞ We would prefer to view all values of classical exp(−x2) dx −∞ special functions of mathematical physics [54] at √ algebraic arguments as being closed forms. Again as π. there is no generally accepted class of special —Collins Dictionary functions, but most practitioners would agree that And of course one cares more for a closed form the solutions to the classical second-order algebraic when the object under study is important, such as differential equations (linear or, say, Painlevé) are when it engages the normal distribution as above. included. But so are various hypertranscendental With that selection recorded, let us turn to some functions, such as Γ ,B, and ζ, which do not arise more formal proposals. in that way.2 Hence, we do not wish to accept any definition Fourth Approach. Various notions of elementary of special function which relies on the underlying numbers have been proposed. functions satisfying natural differential equations. Definition [31]. A subfield F of C is closed The class must be extensible; new special functions under exp and log if (1) exp(x) ∈ F for all are always being discovered. A recent American Mathematical Monthly re- x ∈ F and (2) log(x) ∈ F for all nonzero 3 x ∈ F, where log is the branch of the view of [47] says: natural function such that −π < 2 Im(log x) ≤ π for all x. The field E of EL Of course, a value of a hypertranscendental function numbers is the intersection of all subfields at algebraic argument may be very well behaved; see Example 4. of C that are closed under exp and log. 3Available at http://www.maa.org/maa%20reviews/ —Tim Chow 4221.html.

January 2013 Notices of the AMS 51 There’s no rigorous definition of special functions, but the following definition is in line with the general consensus: functions that are commonly used in applications have many nice properties and are not typically available on a calculator. Obviously this is a sloppy definition, and yet it works fairly well in practice. Most people would agree, for example, that the Gamma function is included in the list of special functions and that trigonometric functions are not. Once again, there is much to agree with and much to quibble about in this reprise. That said, most serious books on the topic are little more spe- cific. For us, special functions are nonelementary (a) modulus of W functions about which a significant literature has developed because of their importance in either mathematical theory or in practice. We certainly prefer that this literature include the existence of excellent algorithms for their computation. This is all consonant with—if somewhat more ecumenical than—Temme’s description in the preface of his lovely book [54, Preface, p. xi]: [W]e call a function “special” when the func- tion, just like the logarithm, the exponential and trigonometric functions (the elemen- tary transcendental functions), belongs to the toolbox of the applied mathematician, the physicist or engineer. Even more economically, Andrews, Askey, and Roy start the preface to their important book (b) W on real line Special Functions [1] by writing: Figure 1. The . Paul Turan once remarked that special functions would be more appropriately labeled “useful functions”. We consider this to be a splendid closed form With little more ado, they then start to work on even though, again assuming Shanuel’s conjecture, the Gamma and Beta functions; indeed, the term W(1) 6∈ E [31]. Additionally, it has only recently “special function” is not in their index. Near the rigorously been proven that W is not an elementary end of their preface, they also write: function in Liouville’s precise sense [26]. We also [W]e suggest that the day of formulas may note that successful simplification in a modern be experiencing a new dawn. CAS [29] requires a great deal of knowledge of special functions.  Example 1 (Lambert’s W). The Lambert W func- tion, W (x), is defined by appropriate solution of Further Approaches y y ·e = x [21, pp. 277–279]. This function has been Fifth Approach. PlanetMath’s offering, as of Feb- implemented in computer algebra systems (CAS) ruary 15, 2010,4 is certainly in the elementary and has many uses despite being unknown to most number corner. scientists and only relatively recently named [41]. expressible in closed form (Definition) An It is now embedded as a primitive in Maple and expression is expressible in a closed form Mathematica with the same status as any other if it can be converted (simplified) into well-studied special or elementary function. (See, an expression containing only elementary for example, the tome [26].) The CAS know its functions, combined by a finite amount of power and much more. For instance, in Maple rational operations and compositions. entering: —PlanetMath > fsolve(exp(x)*x=1);identify(%); returns 4Available at http://planetmath.org/encyclopedia/ 0.5671432904, LambertW(1) ClosedForm4.html.

52 Notices of the AMS Volume 60, Number 1 This reflects both much of what is best and Now we are prepared to posit what is worst about “the mathematical wisdom Definition [7]. The ring of hyperclosure H is of crowds”. For the reasons adduced above, we the smallest subring of C containing the set wish to distinguish—but admit both—those closed X. Elements of H are deemed hyperclosed. forms that give analytic insight from those which In other words, the ring H is generated by all are sufficient and prerequisite for effective com- general hypergeometric evaluations under the ·, + putation. Our own current preferred class [7] is operators, all symbolized by described next. H = hXi·,+ . Sixth Approach. We wish to establish a set X of H will contain a great many interesting closed generalized hypergeometric evaluations; see [7] for forms from modern research. Note that H contains an initial, rudimentary definition which we shall all closed forms in the sense of Wilf and Zeilberger refine presently. First, we want any convergent [48, Ch. 8], wherein only finite linear combinations sum of hypergeometric evaluations are allowed. X n (1) x = cnz So what numbers are in the ring H? First off, n≥0 almost no complex numbers belong to this ring! to be an element of our set X, where z is algebraic, This is easily seen by noting that the set of general c0 is rational, and for n > 0, hypergeometric evaluations is countable, so the A(n) generated ring must also be countable. Still, a c = c − n B(n) n 1 great many fundamental numbers are provably hyperclosed. Examples follow, in which we let ω for integer A, B with deg A ≤ deg B. Un- denote an arbitrary algebraic number and n any der these conditions the expansion for x converges positive integer: absolutely on the open disk |z| < 1. However, we also allow x to be any finite analytic-continuation ω, log ω, eω, π; value of such a series. Moreover, when z lies on the dilogarithmic combination a branch cut, we presume both branch limits to  1  be elements of X. (See ensuing examples for some Li2 √ + (log 2)(log 3); 5 clarification.) It is important to note that our set the elliptic integral K(ω); X is closed under rational due to freedom of choice for c0. the zeta function values ζ(n); Example 2 (First members of X). The generalized special functions such as hypergeometric function evaluation the Bessel evaluations Jn(ω). ! a1, . . . ap+1 Incidentally, it occurs in some modern experi- p+1Fp z b1, . . . , bp mental developments that the real or imaginary part of a hypergeometric evaluation is under scrutiny. for rational a , b with all b positive has branch i j j Generally, <, = operations preserve hyperclosure cut z ∈ (0, ∞), and the evaluation is an element of simply because the series (or continuations) at z X for complex z not on the cut (and the evaluation and z∗ can be linearly combined in the ring H. Re- on each side of said cut is also in X). P n 3 ferring to Example 2, we see that for algebraic z, the The trilogarithm Li3(z) := n≥1 z /n offers a number < (Li3(z)) is hyperclosed, and even on the canonical instance. Formally, 2 < = 7 + π   cut, (Li3(2)) 16 ζ(3) 8 log 2 is hyperclosed. 1 1, 1, 1, 1    Li (z) = F z , ··· 3 4 3 In general, < F z is hyperclosed. z 2, 2, 2 p+1 p ··· and for z = 1/2 the hypergeometric series con- We are not claiming that hyperclosure is any verges absolutely, with kind of final definition for “closed forms”, but we do believe that any defining paradigm for closed  1  7 1 π 2 Li = ζ(3) + log3 2 − log 2. forms must include this ring of hyperclosure H. 3 2 8 6 12 One way to reach further is to define a ring of = Continuation values at z 2 on the branch cut can superclosure as the closure be inferred as S := hHHi . 7 π 2 π ·,+ lim Li3 (2 ± i) = ζ(3) + log 2 ± i log 2, →0+ 16 8 4 This ring contains numbers such as so both complex numbers on the right here are el- 1   eπ + π e, , 1 ζ(3)ζ(5) ements of X. The quadralogarithmic value Li4 2 is thought not to be similarly decomposable but and of course a vast collection of numbers that may likewise belongs to X.  not belong to H itself. If we say that an element of

January 2013 Notices of the AMS 53 S is superclosed, we still preserve the countability is an L-series value over an elliptic curve E with of all superclosed numbers. Again, any good conductor 15. Rogers [50] recasts (2) as definition of “closed form” should incorporate ∞ !2 2n+1 ? 15 X 2n (1/16) whatever is in the ring S. (3) F(3, 5) = , π 2 n 2n + 1 n=0 Seventh Approach. In a more algebraic topological setting, it might make sense to define closed where forms to be those arising as periods, that is, F(b, c) := (1 + b)(1 + c) X (−1)n+m+j+k as integrals of rational functions (with integer × 2 2 2 22 parameters) in n variables over domains defined n,m,j,k (6n + 1) + b(6m + 1) + c(6j + 1) + bc(6k + 1) by algebraic equations. These ideas originate in is a four-dimensional lattice sum. the theory of elliptic and abelian integrals and While (3) remains a conjecture,5 Rogers is able to are deeply studied [42]. Periods form an algebra evaluate many values of F(b, c) in terms of Meijer- and certainly capture many constants. They are G or hypergeometric functions. We shall consider especially well suited to the study of L-series, the most famous crystal sum, the Madelung con- multizeta values, , and the like but stant, in Example 15. again will not capture all that we wish. For example,  e is conjectured not to be a period, as is Euler’s It is striking how beautiful combinatorial games constant γ (see the section “Profound Curiosities”). can be when played under the rubric of hyper- or Moreover, while many periods have nice series, it superclosure. is not clear that all do. As this takes us well outside our domain of Example 4 (Superclosure of Γ at rational argu- expertise, we content ourselves with two examples ments). Let us begin with the Beta function originating in the study of Mahler measures. We (r) (s) B(r, s) := Γ Γ , refer to a fundamental paper by Deninger [39] and Γ (r + s) a very recent paper of Rogers [50] for details. with Γ (s) defined, if one wishes, as Γ (s) := R ∞ s−1 −t Example 3 (Periods and Mahler measures [39]). 0 t e dt. It turns out that for any rationals r, s The logarithmic Mahler measure of a the Beta function is hyperclosed. This is immediate P in n-variables can be defined as from the hypergeometric identities 1 1 1   Z Z Z   1 rs −r, −s 2πiθ1 2πiθn = µ(P) := ··· log P e , . . . , e 2F1 1 , 0 0 0 B(r, s) r + s 1 × dθ ··· dθ . π sin π(r + s) (1 − r)M (1 − s)M 1 n B(r, s) = sin πr sin πs M!(1 − r − s)M Then µ(P) turns out to be an example of a period,   r, s and its exponential, M(P) := exp(µ(P)), is a mean × F 1 , 2 1 of the values of P on the unit n-torus. When M + 1 n = 1 and P has integer coefficients, M(P) is where M is any integer chosen such that the hy- always an algebraic integer. An excellent online pergeometric series converges, say M = d1 + r + se. synopsis can be found in Dave Boyd’s article, (Each of these Beta relations is a variant of the cele- http://eom.springer.de/m/m120070.htm. In- brated Gauss evaluation of 2F1 at 1 [1], [54] and is deed, Boyd has been one of the driving forces in also the reason B is a period.) the field. A brief introduction to the univariate We did not seize upon the Beta function arbitrar- case is also given in [22, pp. 358–359]. ily, for, remarkably, the hyperclosure of B(r, s)±1 There is a remarkable series of recent results— leads to compelling results on the Gamma function many more discovered experimentally than itself. Indeed, consider for example this product proven—expressing various multidimensional of four Beta-function evaluations: µ(P) as arithmetic quantities. Boyd observes that (1/5) (1/5) (2/5) (1/5) there appears to be a tight connection to K-theory. Γ Γ · Γ Γ (2/5) (3/5) An early result due to Smyth (see [51], also [53]) is Γ Γ 0 Γ (3/5)Γ (1/5) Γ (4/5)Γ (1/5) that µ(1 + x + y) = L3(−1). Here L3 is the Dirichlet · · . L-series modulo three. A partner result of Smyth’s Γ (4/5) Γ (5/5) is that µ(1 + x + y + z) = 7 ζ(3)/π 2, a number We know this product is hyperclosed. But upon 5 that is certainly hyperclosed since both ζ(3) and inspection we see that the product is just Γ (1/5). 1/π are. A conjecture of Deninger [39], confirmed to over fifty places, is that 5Equation (2) is now proven, see M. Rogers and W. Zudilin, ? 15 “On the Mahler measure of 1 + X + 1/X + Y + 1/Y ”, preprint (2) µ(1 + x + y + 1/x + 1/y) = L (2) π 2 E (2011), http://arxiv.org/abs/1102.1153.

54 Notices of the AMS Volume 60, Number 1 Along such lines one can prove that for any posi- disk, convergence is geometric, requiring O(D1+) tive rational a/b (in lowest terms), we have hyper- operations to achieve D good digits. However, in closure of powers of the Gamma function in the very many cases this can be genuinely enhanced form to O(D1/2+) [22]. ±b Γ (a/b) ∈ H. Perforce, we therefore have a superclosure result Closed Forms: Why We Care for any Γ (rational) and its reciprocal: In many optimization problems, simple, approximate solutions are more useful than ±1(a/b) ∈ S. Γ complex exact solutions. Again, like calculations show that b(a/b) is a pe- Γ —Steve Wright riod [42]. One fundamental consequence is thus   As Steve Wright observed in a recent lecture on −2 1 = 1 is hyperclosed; thus every integer Γ 2 π sparse optimization, it may well be that a compli- power of π is hyperclosed. cated analytic solution is practically intractable, Incidentally, deeper combinatorial analysis but a simplifying assumption leads to a very practi- 5  1  shows that in spite of our Γ 5 Beta-chain above, cal closed form approximation (e.g., in compressed it really takes only logarithmically many (i.e., sensing). In to appealing to Occam’s razor, O(log b)) hypergeometric evaluations to write Wright instances that: Gamma-powers. For example, (a) the data quality may not justify exactness,   1 1 !4 (b) the simple solution may be more robust, − 1 1 − , − 7 = F 7 7 1 Γ 7 2376 2 1 1 (c) it may be easier to explain/actuate/ implement/store, 2 2 !2 4 4 ! − , − − , − (d) and it may conform better to prior knowl- × F 7 7 1 F 7 7 1 . 2 1 1 2 1 1 edge. As mathematical discovery more and more We note also that for Γ (n/24) with n integer, elliptic integral algorithms are known that converge as fast involves extensive computation, the premium on having a closed form increases. The insight as those for π [27], [22].  provided by discovering a closed form ideally The above remarks on superclosure of Γ (a/b) comes at the top of the list, but efficiency of lead to the property of superclosure for special computation will run a good second. functions such as Jν (ω) for algebraic ω and rational ν and for many of the mighty Meijer- Example 5 (The amplitude of a pendulum). 6 G functions, as the latter can frequently be Wikipedia, after giving the classical small angle written by Slater’s theorem [15] as superpositions (simple harmonic) approximation s of hypergeometric evaluations with composite- L gamma products as coefficients. (See Example 8 p ≈ 2π g for instances of Meijer-G in current research.) There is an interesting alternative way to en- for the period p of a pendulum of length L and vision hyperclosure or at least something very amplitude α, develops the exact solution in a form close to our above definition. This is an idea equivalent to s of J. Carette [28] to the effect that solutions at L  α  algebraic end-points and algebraic initial points p = 4 K sin g 2 for holonomic ODEs—i.e., differential-equation sys- tems with integer-polynomial coefficients—could and then says: be considered closed. One might say diffeoclosed. This integral cannot be evaluated in terms An example of a diffeoclosed number is J1(1), i.e., of elementary functions. It can be rewrit- from the Bessel differential equation for J1(z) with ten in the form of the elliptic function of z ∈ [0, 1]; it suffices without loss of generality the first kind (also see Jacobi’s elliptic func- to consider topologically clean trajectories of the tions), which gives little advantage since over [0, 1]. There is a formal ring of that form is also insoluble. diffeoclosure, which ring is very similar to our H; True, an elliptic-integral solution is not elementary, however, there is the caution that trajectory solu- yet the notion of insolubility is misleading for two tions can sometimes have nontrivial topology, so reasons: First, it is known that for some special precise ring definitions would need to be effected angles α the pendulum period can be given a closed carefully. form. As discussed in [33], one exact solution is, for It is natural to ask, “what is the complexity of hypergeometric evaluations?” Certainly for the 6Available at http://en.wikipedia.org/wiki/ converging forms with variable z on the open unit Pendulum_(mathematics).

January 2013 Notices of the AMS 55 α = π/2 (so pendulum is released from horizontal- these computations are performed using conven- rod position), tional double precision (64-bit IEEE) arithmetic. s ! √ L π Then if a particular phase space point is deemed p = 2π . numerically unstable, it is recomputed with double- g 2(3/4) Γ double precision. These researchers expect that It is readily measurable in even a rudimentary√ lab- further optimization of the procedure for identify- −2 oratory that the excess factor here, πΓ (3/4) ≈ ing unstable points may be required to arrive at an 1.18034, looks just right; i.e., a horizontal-release optimal compromise between numerical accuracy pendulum takes 18 percent longer to fall. More- and speed of the code. Thus they plan to incor- over, there is an exact dynamical solution for the porate arbitrary precision arithmetic into these time-dependent angle α(t), namely, for a pendu- calculations. Their objective is to design a pro- ±∞ = ± = lum with α( ) π and α(0) 0; i.e., the bob cedure where instead of using fixed double or crosses angle zero (hanging straight down) at time quadruple precision for unstable points, the num- zero, but in the limits of time → ±∞ the bob ends ber of digits in the higher precision calculation is up straight vertical. We have period p = ∞, yet the dynamically set according to the instability of the exact angle α(t) for given t can be written in terms point. Any subroutine which uses a closed form of elementary functions! symbolic solution (exact or approximate) is likely The second misleading aspect is this: K is, for to prove much more robust and efficient. any α, remarkably tractable in a computational  sense. Indeed K admits a quadratic transformation √ Detailed Examples 1 − 1 − k2 (4) K (k) = (1 + k1) K (k1) , k1 := √ , We start with three examples originating in [16]. 1 + 1 − k2 In the January 2002 issue of SIAM News, Nick as was known already to Landen, Legendre, and Trefethen presented ten diverse problems used Gauss. in teaching modern graduate numerical analysis In fact all elementary functions to very high students at Oxford University, the answer to precision are well computed via K [22]. So the com- each being a certain real number. Readers were ment was roughly accurate in the world of slide challenged to compute ten digits of each answer, rules or pocket calculators; it is misleading today if with a $100 prize to the best entrant. Trefethen one has access to any computer package. Neverthe- wrote, less, both deserve to be called closed forms: one exact and the other an elegant approximate closed If anyone gets 50 digits in total, I will be form (excellent in its domain of applicability, much impressed. as with Newtonian mechanics) that is equivalent to To his surprise, a total of ninety-four teams,  α  π representing twenty-five different nations, sub- K sin ≈ 2 2 mitted results. Twenty of these teams received for small initial amplitude α. To compute a full one hundred points (ten correct digits for K(π/6) = 1.699075885 ... to five places re- each problem). The problems and solutions are quires using (4) only twice and then estimating the dissected most entertainingly in [16]. One of the resultant integral by π/2. A third step gives the current authors wrote the following in a review [19] of [16]: ten-digit precision shown.  Success in solving these problems required It is now the case that much mathematical computation is hybrid: mixing numeric and sym- a broad knowledge of mathematics and nu- bolic computation. Indeed, which is which may not merical analysis, together with significant be clear to the user if, say, numeric techniques computational effort, to obtain solutions have been used to return a symbolic answer or if and ensure correctness of the results. As a symbolic closed form has been used to make described in [16] the strengths and lim- possible a . Moving from itations of Maple, Mathematica, Matlab classical to modern physics, both understand- (The 3Ms), and other software tools such ing and effectiveness frequently demand hybrid as PARI or GAP, were strikingly revealed in computation. these ventures. Almost all of the solvers relied in large part on one or more of these Example 6 (Scattering amplitudes [2]). An interna- three packages, and while most solvers tional team of physicists, in preparation for the attempted to confirm their results, there Large Hadron Collider (LHC), is computing scat- was no explicit requirement for proofs to tering amplitudes involving quarks, gluons, and be provided. gauge vector bosons in order to predict what re- sults could be expected on the LHC. By default, Example 7 (Trefethen problem #2 [16], [19]).

56 Notices of the AMS Volume 60, Number 1 A photon moving at speed 1 in the x-y plane 4 starts at t = 0 at (x, y) = (1/2, 1/10) head- ing due east. Around every integer lattice 3 point (i, j) in the plane, a circular mirror of 2 radius 1/3 has been erected. How far from 1 the origin is the photon at t = 10? Using interval arithmetic with starting intervals 6 4 2 2 1 of size smaller than 10−5000, one can actually find the position of the particle at time 2000, not just 2 at time ten. This makes a fine exercise in very high- 3 precision interval computation, but in the absence (a) W3 of any closed form, one is driven to such numerical gymnastics to deal with error propagation.  4 Example 8 (Trefethen’s problem #9 [16], [19]). 3 R 2 α 2 The integral I(a) = 0 [2 + sin(10α)]x × sin(α/(2 − x)) dx depends on the param- 1 eter α. What is the value α ∈ [0, 5] at which 6 4 2 2 I(α) achieves its maximum? 1 The maximum parameter is expressible in terms 2 of a Meijer-G function, which is a special func- tion with a solid history. While knowledge of this 3 function was not common among the contestants, (b) W4 Mathematica and Maple both will figure this out Figure 2. Moments of n-step walks in the plane. [15], and then the help files or a Web search will W3 , W4 analytically continued to the real line. quickly inform the scientist. This is another measure of the changing envi- ronment. It is usually a good idea—and not at all Figure 2 and in the in Figure 3. immoral—to data-mine. These Meijer-G functions, The latter highlights the utility of the Meijer-G first introduced in 1936, also occur in quantum representations. Note the poles and removable field theory and many other places [8]. For exam- singularities. ple, the moments of an n-step random walk in the The Meijer-G functions are now described in the plane are given for s > 0 by newly completed Digital Library of Mathematical s 7 Z n Functions and as such are now full, indeed central, X 2πxki (5) Wn(s) := e dx. members of the family of special functions. n  [0,1] k=1 Example 9 (Trefethen’s problem #10 [16], [19]). It transpires [24], [36] that for all complex s A particle at the center of a 10 × 1 rectangle (6) ! undergoes Brownian motion (i.e., 2-D ran- Γ (1 + s/2) 2,1 1, 1, 1 1 W3(s) = G . dom walk with infinitesimal step lengths) (1/2) (−s/2) 3,3 1 , − s , − s 4 Γ Γ 2 2 2 till it hits the boundary. What is the proba- Moreover, for s not an odd integer, we have bility that it hits at one of the ends rather     !2 1 1 1 than at one of the sides? 1 πs s 2 , 2 , 2 1 W (s) = tan F   3 22s+1 2 s−1 3 2 s+3 s+3 4 Hitting the Ends. Bornemann [16] starts his remark- 2 2 , 2 able solution by exploring Monte-Carlo methods, ! s s s ! s − 2 , − 2 , − 2 1 + F . which are shown to be impracticable. He then s 3 2 s−1 4 2 1, − 2 reformulates the problem deterministically as We have not given the somewhat technical defini- the value at the center of a 10 × 1 rectangle of an tion of MeijerG, but Maple, Mathematica, Google appropriate harmonic measure [57] of the ends, searches, Wikipedia, the DLMF, or many other tools arising from a five-point discretization of Laplace’s will. equation with Dirichlet boundary conditions. This is then solved by a well-chosen sparse Cholesky There are two corresponding formulae for W4. We thus know from our “Sixth Approach” section, 7A massive revision of Abramowitz and Stegun, with in regard to superclosure of -evaluations, that Γ the now redundant tables removed, is available at both W3(q), W4(q) are superclosed for rational ar- http://dlmf.nist.gov. The hard copy version is gument q for q not an odd integer. We illustrate also now out [45]. It is not entirely a substitute for the by showing graphs of W3,W4 on the real line in original version, as coverage has changed.

January 2013 Notices of the AMS 57 answer is 2 (9) p = arcsin (k ), π 100 where  √   √   √  k100 := 3 − 2 2 2 + 5 −3 + 10

 √ √ 22 × − 2 + 4 5 .

No one (except harmonic analysts perhaps) antici- pated a closed form, let alone one like this.

Where Does This Come From? In fact, [22, (3.2.29)] shows that ∞ X (−1)n  π(2n + 1)  1 (a) W3 (10) sech ρ = arcsin k 2n + 1 2 2 n=0

exactly when kρ2 is parameterized by theta func- tions in terms of the so-called nome, q = exp(−πρ), as Jacobi discovered. We have 2 P∞ (n+1/2)2 θ2 (q) n=−∞ q −πρ (11) k 2 = = , q :=e . ρ 2 P∞ n2 θ3 (q) n=−∞ q Comparing (10) and (7) we see that the solution is −7 k100 = 6.02806910155971082882540712292 ... · 10 , as asserted in (9). The explicit form now follows from classical nineteenth-century theory, as discussed, say, in [16], [22]. In fact, k210 is the singular value sent by Ramanujan to Hardy in his famous letter of (b) W4 introduction√ [21], [22]. If Trefethen√ had asked√ for a 210 × 1 box or, even better, a 15 × 14 one, W W Figure 3. W3 via (6) and W4 in the complex plane. this would have shown up in the answer, since in general 2  solver. At this point a reliable numerical value of (12) p(a, b) = arcsin ka2/b2 . π 3.837587979 · 10−7 is obtained, and the posed Alternatively, armed only with the knowledge problem is solved numerically to the requisite ten that the singular values of rational parameters places. are always algebraic, we may finish entirely This is the warm-up. We may proceed to develop computationally as described in [19]. two analytic solutions, the first using separation  We finish this section with two attractive ap- of variables on the underlying PDE on a general plied examples from optics and astrophysics × 2a 2b rectangle. We learn that respectively. ∞ 4 X (−1)n  π(2n + 1)  (7) p(a, b) = sech ρ , Example 10 (Mirages [46]). In [46] the authors, π 2n + 1 2 n=0 using geometric methods, develop an exact but im- plicit formula for the path followed by a light ray where ρ := a/b. A second method using conformal propagating over the earth with radial variations in mappings yields the refractive index. By suitably simplifying, they π   (8) arccot ρ = p(a, b) + arg K eip(a,b)π , are able to provide an explicit integral closed form. 2 They then expand it asymptotically. This is done where K is again the complete elliptic integral of with the knowledge that the approximation is good the first kind. It will not be apparent to a reader to six or seven places, more than enough to use it unfamiliar with inversion of elliptic integrals that on optically realistic scales. Moreover, in the case (7) and (8) encode the same solution—though they of quadratic or linear refractive indices, these steps must, as the solution is unique in (0, 1)—and each may be done analytically. can now be used to solve for ρ = 10 to arbitrary In other words, as advanced by Wright, a precision. Bornemann ultimately shows that the tractable and elegant approximate closed form is

58 Notices of the AMS Volume 60, Number 1 trajectories for which indices n being respectively 1 (14) χ (r) = − r 3 + r, 0 6 (15) χ1(r) = sin r, r (16) χ5(r) = p . 1 + r 2/3

The respective√ star radii are thus closed forms z0 = 6 and z1 = π, while for (16), with index n = 5 we have infinite star radius (no positive zero for the pressure χ5). In the spirit of our previous optics example, the Lane-Emden equation is a simplification of (a) A superior mirage a complicated underlying theory—in this astro- physics case, hydrodynamics—and one is rewarded by some closed-form star radii. But what about, say, index n = 2? We do not know a closed-form func- tion for the χ trajectory in any convenient sense. What the present authors have calculated (in 2005) is the n = 2 star radius, as a high-precision number

z2 = 4.352874595946124676973570061526142628112365363213008835302151 ....

If only we could gain a closed form for this special radius, we might be able to guess the nature of the whole trajectory!  (b) An inferior mirage (Photo © Ctein.) Recent Examples Relating to Our Own Work Figure 4. Two impressive mirages. Example 12 (Ising integrals [5], [8]). We recently studied the following classes of integrals [5]. The obtained to replace a problematic exact solution. Dn integrals arise in the Ising model of mathemat- From these forms interesting qualitative conse- ical physics (showing ferromagnetic temperature- quences follow. With a quadratic index, images driven phase shifts; see Figure 5 and [32]), and the are uniformly magnified in the vertical direction; Cn have tight connections to quantum field theory only with higher-order indices can nonuniform [8]: vertical distortion occur. This sort of knowledge 4 Z ∞ Z ∞ 1 du du = ··· 1 ··· n Cn 2 , allows one, for example, to correct distortions of n! 0 0 Pn  u1 un j=1(uj + 1/uj ) photographic images as in Figure 4 with confidence  2 Q ui −uj and efficiency. ∞ ∞  4 Z Z i

2 k − d χ Q (13) r n 1 = −χn, where (in the last line) uk = i=1 ti. dr 2 Needless to say, evaluating these multidimen- with boundary conditions χ(0) = 0, χ0(0) = 1, and sional integrals to high precision presents a daunt- positive real constant n, all of this giving rise to ing computational challenge. Fortunately, in the a unique trajectory χn(r) on r ∈ [0, ∞). (Some au- first case, the Cn integrals can be written as one- thors invoke the substitution χ(r) := rθ(r) to get dimensional integrals: an equivalent ODE for temperature θ; see [30].) The 2n Z ∞ beautiful thing is, where this pressure trajectory C = pKn(p) dp, n n! 0 crosses zero for positive radius r is supposed to 0 be the star radius ; call that zero zn. where K0 is the modified Bessel function. After com- Amazingly, the Lane-Emden equation has puting Cn to 1000-digit accuracy for various n, we known exact solutions for n = 0, 1, 5, the pressure were able to identify the first few instances of Cn

January 2013 Notices of the AMS 59 numerical value as −2γ lim Cn = 2e , n→∞ where γ is the Euler constant; see the section “Pro- found Curiosities”. We later were able to prove this fact—this is merely the first term of an asymptotic expansion—and thus showed that the Cn integrals are fundamental in this context [5]. The integrals Dn and En are much more diffi- cult to evaluate, since they are not reducible to one-dimensional integrals (as far as we can tell); but with certain symmetry transformations and we were able to symbolically reduce the dimension in each case by one or two. In the case of D5 and E5, the resulting 3-D inte- (a) Critical temperature grands are extremely complicated (see Figure 6), but we were nonetheless able to numerically evalu- ate these to at least 240-digit precision on a highly parallel computer system. This would have been impossible without the symbolic reduction. We give the integral in extenso to show the difference between a humanly accessible answer and one a computer finds useful. In this way, we produced the following evalua- tions, all of which, except the last, we subsequently were able to prove:

D2 = 1/3, 2 D3 = 8 + 4π /3 − 27 L−3(2), 2 D4 = 4π /9 − 1/6 − 7ζ(3)/2,

E2 = 6 − 8 log 2, 2 2 E3 = 10 − 2π − 8 log 2 + 32 log 2, E = 22 − 82ζ(3) − 24 log 2 + 176 log2 2, (b) Wolfram Player Demonstration 4 − 256(log3 2)/3 + 16π 2 log 2 − 22π 2/3, Figure 5. The 2-dimensional Ising Model of Ferromagnetism (a) plotting magnetization C and (blue, with peak) and specific heat M (red, (17) decaying) per site against absolute temperature ? E = 42 − 1984 Li (1/2) + 189π 4/10 − 74ζ(3) T (image provided by Jacques Perk) [44, 5 4 pp. 91–93, 245]. − 1272ζ(3) log 2 + 40π 2 log2 2 − 62π 2/3 + 40(π 2 log 2)/3 + 88 log4 2 2 in terms of well-known constants, e.g., + 464 log 2 − 40 log 2, X  1 1  where Li denotes the function. C = L− (2) := − , 3 3 (3n + 1)2 (3n + 2)2 In the case of D2,D3, and D4, these are confir- n≥0 mations of known results. We tried but failed to 7 recognize D in terms of similar constants (the C4 = ζ(3), 5 12 500-digit numerical value is accessible9 if anyone where ζ denotes the Riemann zeta function. When wishes to try to find a closed form or, in the manner we computed Cn for fairly large n, for instance, of the hard sciences, to confirm our data values).

C1024 = 0.63047350337438679612204019271087890435458707871273234 ..., The conjectured identity shown here for E5 was confirmed to 240-digit accuracy, which is 180 digits we found that these values rather quickly ap- beyond the level that could reasonably be ascribed proached a limit. By using the new edition of the to numerical round-off error. Thus we are quite Inverse Symbolic Calculator,8 we identified this 9Available at http://crd.lbl.gov/~dhbailey/ 8Available at http://isc2.carma.newcastle.edu.au/. dhbpapers/ising-data.pdf.

60 Notices of the AMS Volume 60, Number 1 Z 1 Z 1 Z 1 h 2 2 2 2 2 2 with experimental estimate c1 = 0.605443657 ... . E5 = 2(1 − x) (1 − y) (1 − xy) (1 − z) (1 − yz) (1 − xyz) 0 0 0  h    In [6] we were able to derive the exact theoretical − 4(x + 1)(xy + 1) log(2) y5z3x7 −y4z2(4(y + 1)z + 3)x6 − y3z y2 + 1 z2 value of this “QRS constant” c1 as the unique zero 5 2  3  2  2  4 +4(y + 1)z + 5) x + y 4y(y + 1)z + 3 y + 1 z + 4(y + 1)z − 1 x of the Hurwitz zeta ζ(1/2, z/2) on z ∈ (0, 2). In         +y z z2 + 4z + 5 y2 + 4 z2 + 1 y + 5z + 4 x3 + −3z2 − 4z + 1 y2 so doing, we were able to prove the conjectured i h i −4zy + 1) x2 − (y(5z + 4) + 4)x − 1 / (x − 1)3(xy − 1)3(xyz − 1)3 behavior. Moreover, we were able to sketch the h   + 3(y − 1)2y4(z − 1)2z2(yz − 1)2x6 + 2y3z 3(z − 1)2z3y5 + z2 5z3 + 3z2 higher-order asymptotic behavior, something that    +3z + 5) y4 + (z − 1)2z 5z2 + 16z + 5 y3 + 3z5 + 3z4 − 22z3 − 22z2 would have been impossible without discovery of    +3z + 3) y2 + 3 −2z4 + z3 + 2z2 + z − 2 y + 3z3 + 5z2 + 5z + 3 x5 an analytic formula.     + y2 7(z − 1)2z4y6 − 2z3 z3 + 15z2 + 15z + 1 y5 + 2z2 −21z4 + 6z3 Does this deserve to be called a closed form?     +14z2 + 6z − 21 y4 − 2z z5 − 6z4 − 27z3 − 27z2 − 6z + 1 y3 + 7z6 In our opinion, resoundingly yes unless all inverse   functions such as that in Bornemann’s (12) are to be −30z5 + 28z4 + 54z3 + 28z2 − 30z + 7 y2 − 2 7z5 + 15z4 − 6z3 − 6z2 + 15z     eschewed. Such constants are especially interesting +7) y + 7z4 − 2z3 − 42z2 − 2z + 7 x4 − 2y z3 z3 − 9z2 − 9z + 1 y6    in light of even more recent work by Steve Strogatz + z2 7z4 − 14z3 − 18z2 − 14z + 7 y5 + z 7z5 + 14z4 + 3z3 + 3z2 + 14z and his collaborators on chimera, coupled systems 4  6 5 4 3 2  3  5 4 3 +7) y + z − 14z + 3z + 84z + 3z − 14z + 1 y − 3 3z + 6z − z which can self-organize in parts of their domain 2  2  4 3 2  3 2  3 −z + 6z + 3 y − 9z + 14z − 14z + 14z + 9 y + z + 7z + 7z + 1 x and remain disorganized elsewhere; see Figure 7     + z2 11z4 + 6z3 − 66z2 + 6z + 11 y6 + 2z 5z5 + 13z4 − 2z3 − 2z2 + 13z taken from [43]. In this case, observed numerical    +5) y5 + 11z6 + 26z5 + 44z4 − 66z3 + 44z2 + 26z + 11 y4 + 6z5 − 4z4 limits still need to be put in closed form.      −66z3 − 66z2 − 4z + 6 y3 − 2 33z4 + 2z3 − 22z2 + 2z + 33 y2 + 6z3     It is a frequent experience of ours that, as in +26z + 6) y + 11z2 + 10z + 11 x2 − 2 z2 5z3 + 3z2 + 3z + 5 y5     Example 13, the need for high-accuracy computa- + 4 + 3 − 2 + + 4 + 5 + 4 − 3 − 2 + + 3 z 22z 5z 22z 5z 22 y 5z 5z 26z 26z 5z 5 y tion drives the development of effective analytic  4 3 2  2  3 2  2  + 3z − 22z − 26z − 22z + 3 y + 3z + 5z + 5z + 3 y + 5z + 22z + 5 x expressions (closed forms?), which in turn typically 2 2 3 2 4 2  2  + 15z + 2z + 2y(z − 1) (z + 1) + 2y (z − 1) z(z + 1) + y z 15z + 2z + 15 shed substantial light on the subject being studied.   i h +y2 15z4 − 2z3 − 90z2 − 2z + 15 + 15 / (x − 1)2(y − 1)2(xy − 1)2(z − 1)2 i h  Example 14 (Box integrals [3], [7], [23]). There has (yz − 1)2(xyz − 1)2 − 4(x + 1)(y + 1)(yz + 1) −z2y4 + 4z(z + 1)y3        been recent research on the calculation of the ex- + z2 + 1 y2 − 4(z + 1)y + 4x y2 − 1 y2z2 − 1 + x2 z2y4 − 4z(z + 1)y3     i h i pected distance of points inside a hypercube to the − z2 + 1 y2 + 4(z + 1)y + 1 − 1 log(x + 1) / (x − 1)3x(y − 1)3(yz − 1)3 hypercube. Such expectations are also called “box h  2  2  4  2  3 − 4(y + 1)(xy + 1)(z + 1) x z − 4z − 1 y + 4x(x + 1) z − 1 y integrals” [23]. So, for example, the expectation  2   2  2  2  2  i − x + 1 z − 4z − 1 y − 4(x + 1) z − 1 y + z − 4z − 1 log(xy + 1) / h|r~|i for random r~ ∈ [0, 1]3 has the closed form h 3 3 3i h  3 5 7 2 4 x(y − 1) y(xy − 1) (z − 1) − 4(z + 1)(yz + 1) x y z + x y (4x(y + 1) 1 √ 1 1  √        + 5)z6 − xy3 y2 + 1 x2 − 4(y + 1)x − 3 z5 − y2 4y(y + 1)x3 + 5 y2 + 1 x2 3 − π + log 2 + 3 .     4 24 2 +4(y + 1)x + 1) z4 + y y2x3 − 4y(y + 1)x2 − 3 y2 + 1 x − 4(y + 1) z3 Incidentally, box integrals are not just a math-  2 2 2  2  i + 5x y + y + 4x(y + 1)y + 1 z + ((3x + 4)y + 4)z − 1 log(xyz + 1) / ematician’s curiosity; the integrals have been h 3 3 3ii xy(z − 1) z(yz − 1) (xyz − 1) used recently to assess the randomness of brain h i / (x + 1)2(y + 1)2(xy + 1)2(z + 1)2(yz + 1)2(xyz + 1)2 dx dy dz. synapses positioned within a parallelepiped [38]. Indeed, we had cognate results for Z Z Figure 6. The reduced multidimensional integral s ∆d(s) := kx − yk2 dxdy, for E5, which integral has led via [0,1]d [0,1]d extreme-precision numerical quadrature and which gives the moments of the distance between PSLQ to the conjectured closed form given in two points in the hypercube. (17). In a lovely recent paper [52] Stephan Steiner- berger has shown that in the limit, as the dimension goes to infinity, confident in this result, even though we do not  s/p Z Z have a formal proof [5]. 1 s (18) lim kx − ykp dxdy Note that every one of the D,E forms above, d→∞ d [0,1]d [0,1]d !s/p including the conjectured last one, is hyperclosed 2 = in the sense of our “Sixth Approach” section.  (p + 1)(p + 2) Example 13 (Weakly coupling oscillators [49], [6]). for any s, p > 0. In particular, with p = 2, this In an important analysis of coupled Winfree oscil- gives a first-order answer to our earlier published lators, Quinn, Rand, and Strogatz [49] developed request for the asymptotic behavior of ∆d(s). a certain N-oscillator scenario whose bifurca- A quite recent result is that all box integrals tion phase offset φ is implicitly defined, with a h|r~|ni for integer n and dimensions 1, 2, 3, 4, 5 are conjectured asymptotic behavior: sin φ ∼ 1−c1/N, hyperclosed, in the sense of our “Sixth Approach”

January 2013 Notices of the AMS 61 Figure 7. Simulated chimera (figures and parameters from [43]).

section. It turns out that five-dimensional box in- Profound Curiosities tegrals have been especially difficult, depending In our treatment of numbers enjoying hyperclosure on knowledge of a hyperclosed form for a single or superclosure, we admitted that such numbers definite integral J(3), where are countable, and so almost all complex num- Z log(t + x2 + y2) bers cannot be given a closed form along such (19) J(t) := dx dy. 2 2 [0,1]2 (1 + x )(1 + y ) lines. What is stultifying is: How do we identify an A proof of hyperclosure of J(t) for algebraic t ≥ 0 explicit number lying outside such countable sets? is established in [23, Thm. 5.1]. Thus h|r~|−2i for r~ ∈ The situation is tantamount to the modern bind [0, 1]5 can be written in explicit hyperclosed form in regard to normal numbers, numbers which to involving a 105-character symbolic J(3); the au- some base have statistically random digit structure thors of [23] were able to reduce the 5-dimensional in a certain technical sense. The bind is, though 4 box integral down to “only” 10 characters. A com- almost all numbers are absolutely normal (i.e., 5 panion integral J(2) also starts out with about 10 normal to every base 2, 3,... ), we do not know characters but reduces stunningly to only a few a single fundamental constant that is provably dozen characters, namely, absolutely normal. (We do know some “artificial” 2   π 7 11 π normal numbers; see [14].) (20) J(2) = log 2 − ζ(3) + π Cl2 8 48 24 6 Here is one possible way out of the dilemma.   29 5π In the theory of computability, the existence of − π Cl2 , 24 6 noncomputable real numbers, such as an encoded where Cl2 is the Clausen function Cl2(θ) := list of halting Turing machines, is well established. P sin(nθ)/n2 (Cl is the simplest nonelemen- n≥1 2 The celebrated Chaitin constant Ω is a well-known tary Fourier series). noncomputable. So a “folk” argument goes: Since Automating such reductions will require a every element of the ring of hyperclosure H can be sophisticated simplification scheme with a very computed via converging series, it should be that large and extensible knowledge base. With a 6∈ current research assistant, Alex Kaiser at Berke- Ω H. A good research problem would be to make ley, we have started to design software to refine this heuristic rigorous. and automate this process and to run it before Let us focus on some constants that might submission of any equation-rich paper (see [9]). not be hyperclosed (nor superclosed). One This semi-automated integrity checking becomes such constant is the celebrated Euler constant Pn pressing when, as above, verifiable output from γ := limn→∞ k=1 1/k − log n. We know of no a symbolic manipulation can be the length of a hypergeometric form for γ; said constant may well Salinger novella.  lie outside H (or even S). There are expansions for

62 Notices of the AMS Volume 60, Number 1 said crystal is NaCl (salt), as illustrated in Figure 8). Image (b) is of a Helaman Ferguson sculpture based on M which is awarded biannually by the Canadian Mathematical Society as part of the David Borwein Career Award) and is given by the formal (conditionally convergent [18]) sum X (−1)x+y+z (21) M := p x2 + y2 + z2 (x,y,z)6=(0,0,0) = −1.747564594633... and has never been given what a reasonable observer would call a closed form. Nature plays an interesting trick here: There are other crystal structures that are tractable, yet somehow this (a) NaCl nearest neighbors exquisitely symmetrical salt structure remains elusive. In general, even-dimensional crystal sums are more tractable than odd for the same modular function reasons that the number of representa- tions of a number as the sum of an even number of squares is. But this does not make them easy, as illustrated by Example 2. Here we have another example of a constant having no known closed form yet is rapidly calcu- lable. A classical rapid expansion for the Madelung constant is due to Benson: (22) X  π q  M=−12π sech2 (2m + 1)2 +(2n + 1)2 , 2 m,n≥0 in which convergence is exponential. Summing for m, n ≤ 3 produces −1.747564594 ... , correct to (b) CMS Prize Sculpture eight digits. There are a great many other such Figure 8. Two representations of salt. formulae for M (see [22], [35]). Through the analytic methods of Buhler, Cran- dall, Tyagi, and Zucker since 1999 (see [35], [37], the Euler constant, such as [55], [58]), we now know approximations such as   1 3 3 4 X k+1 log(2k + 1) 1 log 2 8π 1 ( ) ( ) k2 γ = log π−4 log Γ + (−1) , M ≈ − − + + √ + Γ 8 Γ√8 +log 4 , 4 π 2k + 1 3/2 0 k≥1 8 4π 3 8 π 2 16k4 1/4 1/4 2 and even more exotic series (see [13]). But in the where k4 := ((2 − 1)/(2 + 1)) . Two remark- spirit of the present treatment, we do not want able things: First, this approximation is good to the to call the infinite series closed, because it is not same thirteen decimals we give in (21); the miss- hypergeometric per se. Relatedly, the classical ing O(10−14) error here is a rapidly, exponentially Bessel expansion is converging—but alas infinite—sum in this modern     ∞ Pn−1 1 2 !n approximation theory. Second, this six-term ap- z X = z K (z)= − ln + γ I (z)+ k 1 k . 0 0 2 proximation itself is indeed hyperclosed, the only 2 = (n!) 4 n 1 problematic term being the Γ -function part, but we Now K0(z) has a (degenerate) Meijer-G did establish in our “Sixth Approach” section that representation—so potentially is superclosed for B(1/8, 3/8) and also 1/π are hyperclosed, which is algebraic z—and I0(z) is accordingly hyperclosed, enough. Moreover, the work of Borwein and Zucker but the nested harmonic series on the right is [27] also settles hyperclosure for that term.  again problematic. Again, γ is conjectured not to Certainly we have nothing like a proof, or even be a period [42]. the beginnings of one, that M (or γ) lies outside H Example 15 (Madelung constant [22], [37], [58]). (or even S), but we ask on an intuitive basis, Is a Another fascinating number is the Madelung con- constant such as the mighty M telling us that it stant, M, of chemistry and physics [22, Section is not hyperclosed, in that our toil only seems to 9.3]. This is the potential energy at the origin of bring more “closed-form” terms into play, with no an oscillating-charge crystal structure (most often exact resolution in sight?

January 2013 Notices of the AMS 63 Concluding Remarks and Open Problems have illustrated this. Moreover, we belong to the • We have posited several approaches to the elusive subset of mathematicians that finds fun in finding notion of “closed form”, but what are the intersec- unanticipated closed forms. tions and interrelations of said approaches? For example, can our “Fourth Approach” be precisely Acknowledgements absorbed into the evidently more general “Sixth Thanks are due to David Bailey and Richard Brent Approach” (hyperclosure and superclosure)? for many relevant conversations and to Armin • How do we find a single number that is provably Straub for the complex plots of W3 and W4. not in the ring of hyperclosure H? (Though no such number is yet known, almost all numbers are, as References noted, not in said ring!) The same question persists [1] G. E. Andrews, R. Askey, and R. Roy, Special Functions, for the ring of superclosure, S. Furthermore, how Cambridge University Press, 1999. precisely can one create a field out of HH via [2] David Bailey, Roberto Barrio, and Jonathan appropriate operator extension? Borwein, High-precision computation: Mathemati- • Though H is a subset of S, how might one prove cal physics and dynamics, Applied Mathematics and Computation 218 (2012), 10106–10121, that 6= ? (Is the inequality even true?) Likewise, H S http://dx.doi.org/10.1016/j.amc.2012.03.087. is the set of closed forms in the sense of [48, Ch. 8] [3] D. Bailey, J. Borwein, N. Calkin, R. Girgensohn, (only finite linear combinations of hypergeometric R. Luke, and V. Moll, Experimental Mathematics in evaluations) properly contained in our H? And Action, A K Peters, Natick, MA, 2007. H what about a construct such as HH ? Should such [4] D. H. Bailey and J. M. Borwein, Computer-assisted an entity be anything really new? Lest one remark discovery and proof, Tapas in Experimental Mathe- on the folly of such constructions, we observe that matics, 21–52, in Contemporary Mathematics, vol. 457, π π American Mathematical Society, Providence, RI, 2008. most everyone would say π is a closed form! [5] D. H. Bailey, D. Borwein, J. M. Borwein, and • Having established the property of hyperclosure R. E. Crandall, Hypergeometric forms for Ising- b for Γ (a/b), are there any cases where the power class integrals, Experimental Mathematics 16 (2007), 257–276. b may be brought down? For√ example, 1/π is hyperclosed, but what about 1/ π? [6] D. H. Bailey, J. M. Borwein, and R. E. Crandall, Resolution of the Quinn–Rand–Strogatz constant • What is a precise connection between the ring of nonlinear physics, Experimental Mathematics 18 of hyperclosure (or superclosure) and the set of (2008), 107–116. periods or of Mahler measures (as in Example 3)? [7] D. H. Bailey, J. M. Borwein, and R. E. Crandall, • There is expounded in reference [23] a theory of Advances in the theory of box integral, Mathematics “expression entropy”, whereby some fundamental of Computation, 79 (2010), 1839–1866. [8] D. H. Bailey, J. M. Borwein, D. M. Broadhurst, and entropy estimate gives the true complexity of L. Glasser, Elliptic integral representation of Bessel an expression. So, for example, an expression moments, J. Phys. A: Math. Theory 41 (2008), 5203– having one thousand instances of the polylog 5231. token Li3 might really involve only about 1,000 [9] D. H. Bailey, J. M. Borwein, and A. Kaiser, Automated characters, with that polylogarithm token encoded simplification of large symbolic expressions, preprint, as a single character, say. (In fact, during the August 2012. research for [23] it was noted that the entropy [10] D. H. Bailey, J. M. Borwein, V. Kapoor, and E. Weis- stein, Ten problems in experimental mathematics, of Maple and Mathematica expressions of the American Mathematical Monthly 113 (2006), 481–409. same entity often had widely varying text-character [11] D. H. Bailey, P. B. Borwein, and S. Plouffe, On counts but similar entropy assessments.) the rapid computation of various polylogarithmic On the other hand, one basic notion of “closed constants, Mathematics of Computation 66 (1997), form” is that explicitly infinite sums not be allowed. 903–913. Can these two concepts be reconciled? Meaning, [12] R. Baillie, D. Borwein, and J. Borwein, Some sinc can we develop a theory of expression entropy sums and integrals, American Math. Monthly 115 (2008), 888–901. by which an explicit, infinite sum is given infinite [13] D. Bailey and R. Crandall, On the random character entropy? This might be difficult, as, for example, of fundamental constant expansions, Experimental P∞ 1 a sum n=1 n3/2 takes only a few characters to Mathematics 10 (2001), 175–190. symbolize, as we just did! If one can succeed, [14] , Random generators and normal number, though, in thus resolving the entropy business for Experimental Mathematics 11 (2002), 527–547. expressions, “closed form” might be rephrased as [15] Folkmar Bornemann, How Mathematica “finite entropy”. and Maple Get Meijer’s G-function into Prob- lem 9, preprint, 2005. Available at http:// In any event, we feel strongly that the value www-m3.ma.tum.de/bornemann/ of closed forms increases as the complexity of Numerikstreifzug/Chapter9/MeijerG.pdf. the objects we manipulate computationally and [16] F. Bornemann, D. Laurie, S. Wagon, and J. Wald- inspect mathematically grows, and we hope we vogel, The SIAM 100-Digit Challenge: A Study

64 Notices of the AMS Volume 60, Number 1 in High-Accuracy Numerical Computing, SIAM, [40] H. R. P. Ferguson, D. H. Bailey, and S. Arno, Anal- Philadelphia, 2004. ysis of PSLQ, an integer relation finding algorithm, [17] E. J. Borowski and J. M. Borwein, Web-Linked Dic- Mathematics of Computation 68, no. 225 (1999), tionary of Mathematics, Smithsonian/Collins Edition, 351–369. 2006. [41] Brian Hayes, Why W? American Scientist 93 (2005), [18] D. Borwein, J. M. Borwein, and K. F. Taylor, Con- 1004–1008. vergence of lattice sums and Madelung’s constant, [42] M. Kontsevich and D. Zagier, Periods. In Mathe- J. Math. Phys. 26 (1985), 2999–3009. matics Unlimited—2001 and Beyond, Springer-Verlag, [19] J. M. Borwein, The SIAM 100 Digits Challenge, ex- 2005, pp. 771–808. tended review in the Mathematical Intelligencer 27 [43] E. A. Martens, C. R. Laing, and S. H. Strogatz, Solv- (2005), 40–48. able model of spiral wave chimeras, Physical Review [20] J. M. Borwein and D. H. Bailey, Mathematics by Ex- Letters 104 (2010), 044101. periment: Plausible Reasoning in the 21st Century, [44] B. McCoy and Tai Tsun Wu, The Two-Dimensional extended second edition, A K Peters, Natick, MA, 2008. Ising Model, Harvard Univ. Press, 1973. [21] J. M. Borwein, D. H. Bailey, and R. Girgensohn, Ex- [45] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, perimentation in Mathematics: Computational Roads and C. W. Clark, NIST Handbook of Mathematical to Discovery, A K Peters, Natick, MA, 2004. Functions, Cambridge University Press, 2010. [22] J. M. Borwein and P. B. Borwein, Pi and the AGM, [46] B. D. Nener, N. Fowkes, and L. Borredon, Analytical John Wiley, 1987. models of optical refraction in the troposphere, JOSA [23] J. Borwein, D. Nuyens, A. Straub, and James Wan, 20 (2003), 867–875. Some arithmetic properties of short random walk [47] K. Oldham, J. Myland, and J. Spanier, An Atlas of integrals, Ramanujan Journal 26 (2011), 109–132. Functions: With Equator, the Atlas Function Calculator, [24] Jonathan M. Borwein, Armin Straub, and James Springer-Verlag, 2009. Wan, Three-step and four-step random walks, [48] M. Petkovsek, H. Wilf, and D. Zeilberger, A = B, Experimental Mathematics, in press, January 2013. A K Peters, Ltd., 1996. Available at http://www.math. [25] J. M. Borwein and K. Devlin, The Computer as upenn.edu/~wilf/AeqB.html. Crucible, A K Peters, Natick, MA, 2008. [49] D. Quinn, R. Rand, and S. Strogatz, Singular un- [26] M. Bronstein, R. M. Corless, J. H. Davenport, and locking transition in the Winfree model of coupled D. J. Jeffrey, Algebraic properties of the Lambert W oscillators, Phys. Rev. E 75 (2007), 036218-1-10. function from a result of Rosenlicht and of Liouville, [50] M. Rogers, Hypergeometric formulas for lattice sums Integral Transforms and Special Functions 19 (2008), and Mahler measures, preprint, 2010. 709–712. [51] C. J. Smyth, Explicit formulas for the Mahler mea- [27] J. M. Borwein and I. J. Zucker, Fast evaluation of sures of families of multivariable polynomials, preprint, the gamma function for small rational fractions using 2003. complete elliptic integrals of the first kind, IMA J. [52] S. Steinerberger, Extremal uniform distribution and Numerical Analysis 12 (1991), 519–526. random chord lengths, Acta Mathematica Hungarica [28] J. Carette, private communication, 2010. 130 (2011), 321–339. [29] , Understanding expression simplification, Pro- [53] C. J. Smyth, The Mahler measure of algebraic num- ceedings of International Conference on Symbolic and bers: A survey, Conference Proceedings, University of Algebraic Computation, ACM, 2004, pp.72–79. Bristol, 3–7 April 2006. LMS Lecture Note Series, vol. [30] S. Chandrasekhar, An Introduction to the Study of 352, Cambridge University Press, Cambridge, 2008, Stellar Structures, Dover, New York, 1967. pp. 322–349. [31] T. Y. Chow, What is a closed-form number? American [54] N. Temme, Special Functions, an Introduction to the Mathematical Monthly 106 (1999), 440–448. Classical Functions of Mathematical Physics, John Wiley [32] B. A. Cipra, An introduction to the Ising model, Amer. & Sons, New York, 1996. Math. Monthly 94 (1987), no. 10, 937–959. [55] S. Tyagi, New series representation for the Madelung [33] R. E. Crandall, Topics in Advanced Scientific constant, Progr. Theoret. Phys. 114 (2005), 517–521. Computation, Springer, New York, 1996. [56] E. W. Weisstein, Closed-Form Solution, [34] , Theory of ROOF Walks, 2007. Available from MathWorld—A Wolfram Web Resource. at http://people.reed.edu/~crandall/papers/ http://mathworld.wolfram.com/Closed- ROOF.pdf. FormSolution.html. [35] , New representations for the Madelung con- [57] B. L. Walden and L. A. Ward, A harmonic mea- stant, Experimental Mathematics 8 (1999), 367–379. sure interpretation of the arithmetic-geometric mean, [36] , Analytic representations for circle-jump mo- American Mathematical Monthly 114 (2007), 610–622. ments, PSIpress, 21 November 2009; http:// [58] I. J. Zucker, Some infinite series of exponential and www.perfscipress.com/papers/ , SIAM J. Math. Anal. 15 (1984), AnalyticWn_psipress.pdf 406–413. [37] R. E. Crandall and J. E. Buhler, Elementary expan- sions for Madelung constants, J. Phys. A: Math. Gen. 20 (1987), 5497–5510. [38] R. Crandall, On the fractal distribution of brain synapses, preprint, 2010. [39] C. Deninger, Deligne periods of mixed motives, theory and the entropy of certain actions, J. Amer. Math. Soc. 10 (1997), 259–281.

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