ISSN 00271349, Moscow University Physics Bulletin, 2010, Vol. 65, No. 1, pp. 6–12. © Allerton Press, Inc., 2010. Published in Russian in Vestnik Moskovskogo Universiteta. Fizika, 2010, No. 1, pp. 8–14. Superfimctions and sqrt of Factorial D. Kouznetsova and H. Trappmannb a Institute for Laser Science, University of ElectroCommunications 151 Chofugaoka, Chofushi, Tokyo, 1828585, Japan b Kameruner Str. 9, 13351 Berlin, Germany email: [email protected], [email protected] Received August 13, 2009 Abstract—The holomorphic function h is constructed such that hhz = z!; this function is interpreted as square root of Factorial. Key words: sqrt of Factorial, superfunction, SuperFactorial, inverse problem. DOI: 10.3103/S0027134910010029 INTRODUCTION 1. Superfunctions This work was motivated by one exercise from the The evaluation of fractional power of a function, pastcentury course of Quantum Mechanics of the i.e., the fractional iteration, (for example, exp , see Moscow State University. It was suggested to give [7–9], or ! ), can be based on the concept of super sense to the operator ! [1]. That time, a satisfactory function [4–6]. For a given function H, which is solution was not found; the opinion was, that such an referred to as the “Transfer Function” below, a super operation has no meaning [2]. function F is a holomorphic solution of the Abel equa tion In Quantum Mechanics, the repeated (iterated) application of an operation (usually, some “observ F(z +1) = H(F(z)). (1) able”) to some argument (which may have sense of Such an equation is pretty old [10–12], although in vector of state) is interpreted as “power” of the opera 1827, N.H. Abel wrote it in a different form, for the tion; in particular, in such a way the square of coordi inverse function of F. The Abel equation comes from nate or the square of momentum are treated. For this the phenomenological consideration of the transfor analogy, the notation without parenthesis is used. In these notations, sinα means sin(α), ln sinz means ln(sin(x)) and so on; such notations are used also in y textbooks on elementary algebra. To avoid confusions 5 at the iterations, we use also the prefix notation Facto 4 rial z = Factorial(z) instead of z!. 3 We assume, that factorial is known meromorphic function, just Gamma function [3] with displaced 2 argument. In this way factorial is interpreted in pro 1 gramming languages Mathematica and Maple. The factorial of the real argument is shown in Fig. 1. 0 x = −1 y In this work, the square root of the Factorial is interpreted as a holomorphic function h such that its −2 second iteration is Factorial, i.e. hhz = z!. For real val −3 ues of argument the graphic of function ! is shown in −4 Fig. 2. Below we describe, how to evaluate it not only for real, but also for complex values of the argument, −5 using superfunctions [4–6]. −6 −6 −5 −4 −3 −2 −1 0 1 2 3 4 x 1 The Englishlanguage version of the article was prepared by the Fig. 1. Factorial of the real argument and graphical solu authors. tion of equation x! = x. 6 SUPERFIMCTIONS AND SQRT OF FACTORIAL 7 mation of a signal F in a singledimensional homoge y ) nous nonlinear system, characterized with the Trans x 10 ) 10 ( 2 (a) x (b) / ( 1 fer Function . Equation (1) may have also other H ) l l 9 a 9 x i a ( i r 2 r applications, discussed below in the special section. In F / o o 1 t t ) c some sense, the Eq. (1) is equivalent of the Schröder’s 8 c x 8 = a a ( y x F l k F equation [13–17]; at the exponential transformation a e i = = r 7 7 + y o y of the argument, the inverse Schröder’s function t 2 c a 6 6 = becomes a superfunction, but not every supefunction F y = can be simply expressed through some inverse 5 y 5 Schröder function. For this reason, here we deal with superfunctions and not with the Schröder functions. 4 4 A superfunction F determines the fractional itera 3 3 tion Hc of the Transfer Function H: 2 2 Hc(z) = F(c + F–1(z)). (2) 1 1 The resulting function Hc can be considered as 0 0 1 2 3 4 5 −2 −1 0 1 2 3 fractional power of function H, because it satisfies the x expected relations H1 = H, Hc + d(z) = HcHdz = Hc(Hd(z)), Fig. 2. Left: y = Factorial(x), thick dashed curve; y = ! (x) = 1/2 1/2 c d c + d Factorial (x), solid curve; y = x! = Factorial(x) , thin i.e., for two numbers c and d, the identity H H = H dashed, as functions of real x; Right: y = F(x), solid, com holds, as if H would be not a function but a number. In pared to y = 2+exp(kx), dashed. particular with c = 1/2, the halfiteration h(z) = H (z) = H1/2(z) is considered to be the square root of Superfunctions of the exponential (see row 4 of the function H, because hhz = h(h(z)) = Hz = H(z). In this Table 1) yet are not so widely known, although Hel sense h2 = H and h = H . muth Kneser had reported the halfiteration of the Some superfunctions (see Table 1) are well known; exponential, i.e., exp , in year 1950 [7]. Here, tetra they are used without to identify them as superfunc tional tet is the superfunction of exp, caracterized in ރ ∈ tions. Several elementary superfunctions (in particu that tetb(0) = 1 and holomorphie on the range \{x lar, trigonometric) were listed also at websites [5, 6, 18]. ޒ : x ≤ 0}; i.e., holomorphie solution of equation Table 1. Examples of superfunctions No. H(z) F(z) F–1(z) Comment 1 z + 1 b + zz – bb ∈ ރ 2 b + zbz + c (z – c)/bb ≠ 0 z c ⎛⎞c 3 bz + c b + log z – 1 – b b⎝⎠1 – b z –1() 4 b tetb(z)tetb z (3), (4), [4, 9, 19–22] b z 5 z exp(b ) log b()ln()z b > 0 6ln(b + ez)ln(bz) ez/bb ≠ 0 7(ab + zb)1/b az1/b (z/a)b a > 0, b ≠ 0 2 z 82z – 1 cos()2 log 2()arccos()z 2 z 92z – 1cosh()2 log 2()arccosh()z compare no. 8 2 z 10 2z/(1 – z ) tan()2 log 2()arctan()z 2 z ⎛⎞⎛⎞z + 1 11 2z/(1 + z ) tanh()2 log 2 2ln ⎝⎠⎝⎠z – 1 12 Factorial(z) SuperFactorial(z) ArcSuperFactorial(z)(6), (8) P(H(Q(z))) P(F(z)) F–1(Q(z)) P(Q(z)) = z MOSCOW UNIVERSITY PHYSICS BULLETIN Vol. 65 No. 1 2010 8 KOUZNETSOV, TRAPPMANN (a)q (b) = y y q 1 . 1 = 2 q 8 . 1 = = 0 .4 6 6 1 q . = 6 6 . q 0 = q = q 1.8 .4 4 4 0 = q p = 0 q = 2 0.2 2 q = 0 2 q = p = 0 0 q = 0 0 Cut q = 0 p = 0 .8 q q = 0 − 2 = − −2 −2 q = 2 0. 0 = 2 0 p 3 = = 0 = p 0 .2 q = .8 p 3 −4 p = 0 −4 −1 q = 0 q = = p = 0 p q 6 q = . q 0 −1 4 p 4 = . = . = 3 − q 0 − = 2 6 6 1 . − = = q − − p 0 1 p 1 = − 0 = . q 8 q = − −8 8 q −6 −4 −20 2 4 6x −6 −4 −20 2 4 6x Fig. 3. f = Factorial(z), left, and f = ArcFactorial(z), right, in the complex zplane; Levels ℜ(f) = p = constant and Levels ᑣ(f)= q = constant are shown with thick lines for p = –4, –3, –2, –1, 0, 1, 2, 3, 4 and for q = –4, –3, –2, –1, 0, 1, 2, 3, 4. tetb(z +1) = expb(tetb(z)). (3) superfunction for the given Transfer Function, namely, Factorial. This allows to evaluate the holo At integer values of z, tetration tetb(z) is result iter ational application of exponential to unity: morphic function ! , giving sense to the logo of the () Physics department of the Moscow State University. tetbz = expb(expb(..expb 1 ..)) (4) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ zrepetitions of exponential Evaluation of tetb at b > exp(1/e) and in particular, 2. Factorial, ArcFactorial and the Fixed Points for b = e and b = 2 is described in [9, 19, 22]. For b = e, For the evaluation of superFactorial and its inverse, the fast approximation is available at [20, 21]. The efficient implementations of Factorial and ArcFacto evaluation at 1 < b < exp(1/e) and, in particular, for b = rial are required. Complex maps of Factorial and Arc 2 , is considered in [4]. Factorial are shown in Fig. 3. In the right hand side of New superfunctions can be obtained by transfor the left picture, the density of levels for the Factorial is mation of the already established superfunctions. If so high that they would overlap; only levels p = 0 and some F is SuperFunction of some Transfer Function q = 0 are shown there. We use the original numerical H, then another superfunctions Ᏺ can be define as C++ implementation [23] of the Factorial and Arc Factorial. Ᏺ(z) = F(z + δ(z)), where δ is some holomorphic 1 periodic function. In particular, the superfunction in At the analysis of a superfunction for same Transfer row 9 can be obtained from that in row 8 with δ(z) = Function H, the key question is about the fixed points πiln(2)/2 = constant.