RE: RE: REPOSTING: PowerTower extended to real exponents

*To*: mathgroup at smc.vnet.net*Subject*: [mg72295] RE: [mg72200] RE: [mg72033] REPOSTING: PowerTower extended to real exponents*From*: "Ingolf Dahl" <ingolf.dahl at telia.com>*Date*: Tue, 19 Dec 2006 06:35:08 -0500 (EST)*Reply-to*: <ingolf.dahl at telia.com>

I think I have to back-off a half step from my claims. But I do not regret what I have written: I think it is a really good math lesson for me, and maybe for other. And I think the conclusions are quite interesting: It is possible to extend PowerTower to real exponents in a consistent way, BUT the solution will not be unique. And that is not by our ignorance, that is by mathematical principle. The case is similar to 0^0, but without a clear "best choice". So we are free to let UN make a law and decide which solution to choose. And such constants as Pi^^E will become somewhat fuzzy, and the value I have published is possible but not necessary. It is not a very common situation in mathematics that constants have intrinsic uncertainness, even if we are used to the fact that the primitive function to an arbitrary function always contains a constant of integration. And in the end of this letter I think I anyway see the opening of the tunnel. What is the problem? One thing that I had assumed was that if f1 is the solution to the equation f[f[x]] == g[x] for g[x] = 2*Sinh[x] and f2 is the solution for g[x] = E^x, we could with very good precision approximate f2 with f1 if x is large enough (say x > 20 or so). It seems reasonable that the relative error we do in that approximation should have an upper bound E^(-2*x). The function f1 can be found by using the series expansion of 2*Sinh[x] for small x, since x=0 is a fixed point of 2*Sinh[x]. But the mathematics is not so kind and well-behaved. If we compare f1 with the corresponding solution f3 for g[x] = Exp[x]-1, we see that they also should have the same asymptotic behaviour. Both are derived from the analytical expression at small x, but anyway the difference between f1 and f3 will show oscillations when x gets large. When we pick out the analytic expression for the solution f1 and f3, these solutions are special and unique just at the fixed point, but a bit away from the fixed point the mathematics do not discern between that special solution and other solutions (which might oscillate near to the fixed point, maybe similar to x*Sin[1/x]). I have obtained a solution f2 for g[x] = E^x by identifying the asymptotic behaviour with that for f1, but I could as well have used f3, and would then obtain another good solution. f2 is thus not an unique solution, even if it seems to be a nice solution. This is also illustrated if we solve the equation f[f[x]] == g[x] for g[x] = Exp[k*x] for small k (0 < k < 1/E). There g[x] has two fixed point, and the smaller one can be used to find the solution for x < E, and the larger fixed point can be used for x > E, by identification of Taylor expansion coefficients. It is thus not needed to compare asymptotes in this case. If we then calculate the function f, we will find that the curves do not connect at x = E. There is a jump of size approx. 0.003 for k=0.01. I plan to include a curve describing this in my next version of the PowerTower notebook on my web site. The PowerTower function is not affected by this jump, since this function will just sample values on the left side of the smaller fixed point. Thus we can anyway use this fixed point to define PowerTower for real exponents for 1 < x <= Exp[1/E]. For larger values of k, there are thus several solutions f(x) available, where it is difficult to say that one is "unique", since there are no fixed points for Exp[k * x]. (Maybe one could find a solution that is the "best", according to some criterion.) Then this spills over to PowerTower too. If x > Exp[1/E], the range of PowerTower[x, n] for real values of n >= -2 is from -Infinity up to (approaching) +Infinity. Then if we would have PowerTower defined in some unique way, the relation PowerTower[y, n] = f[PowerTower[y, n - 1/2]] should define an unique solution f(x) to f[f[x]] == exp[k*x], and thus we get a contradiction if we state that there is no unique solution to this equation. All this does not seem to be common knowledge today (e.g. it is not clear for me from Wikipedia), but much of the behaviour follows directly from the description by G. Szekeres in "Abel's equation and regular growth: Variations of a theme by Abel", see www.emis.de/journals/EM/restricted/7/7.2/szekeres.ps. I have just found it myself (by a tip in a post in http://groups-beta.google.com/group/sci.math.research/tree/browse_frm/thread /42b60356c806380c/016bbf4f08145cfa?rnum=1&hl=en&_done=%2Fgroup%2Fsci.math.re search%2Fbrowse_frm%2Fthread%2F42b60356c806380c%2F016bbf4f08145cfa%3Fhl%3Den %26#doc_e47a95a5c6d3c0f1) and have not yet understood all details there. Anyway, even if the solution f is not unique, there could be useful to have available some nice solutions from the solution set. Then we might at least be able to choose "the best" or "the best known so far". I have solved the equation by letting f[x] ride on the asymptotic behaviour of some "horse function", with fixed points. (I think that this is my contribution to this field of mathematics.) In the paper by Szekeres it is indicated that Exp[k*x] - 1 should display "regular growth" for k > 1. If that is true, it could be the ideal horse function. Another similar alternative, which also might have regular growth behaviour, is Exp[k*x] - (1 + Log[k])/k, which becomes identical to Exp[k*x] at the k value 1/E. So just now it appears to me as the case is almost solved. Are there more small devils hiding in the bush? I have got some nice letters and posts commenting my proposals, but have not had time yet to comment all comments and answer all questions, but I am working at it. Best regards Ingolf Dahl > -----Original Message----- > From: Ingolf Dahl [mailto:ingolf.dahl at telia.com] > Sent: den 14 december 2006 11:49 > To: mathgroup at smc.vnet.net > Subject: [mg72200] REPOSTING:RE: [mg72033] REPOSTING: > PowerTower extended to real exponents > > And then silence... > > I have extended the code to cover x > 1 for PowerTower[x,n], > with some mathematical problems slightly above x = Exp[1/E] = > 1.444667861009766. Since the code has grown (800 kB), I do > not attach it here. You are welcome to download it from > http://web.telia.com/~u31815170/Mathematica/ > I would be grateful if someone could check if my calculations > make sense or are nonsense. I have difficulties to find out > the relevance of this, if it is important or not. It is of > course important for me, because I do this at hobby work, and > want to know if I should apply for means to proceed > professionally with this. > > From http://forum.wolframscience.com/archive/topic/956-1.html: > "Please also note that tetration is used in the proposed RRHC > number notation hyper-format (s=4) and that any positive real > number can be represented as p*(b#n), with this format. > Number p (with 0 < p < b), the super-exponent extension, can > be found by iteratively applying n times the log b operator, > until p < b. In other words, it is not indispensable to solve > the problem of extending tetration to the reals (which has a > great theoretical importance), before using the RRHC number > notation format. > The extension of tetration to the reals is tightly connected > with the analytic continuation of operators, as performed, > for instance, within the "Fractional Calculus". A solution is > probably . around the corner." > > From http://wiki.tetration.org/index.php?title=Main_Page: > "Extending Tetration > Many people are interested in extending tetration from the > natural numbers to the real and complex numbers. Ioannis Galidakis > (http://ioannis.virtualcomposer2000.com/math/) even has an > artcle scheduled for publication that extends research of > tetration into the realm of quaternions. Ioannis also has a > considerable amount of information about tetration on his > website. His growing list of publications shows not only that > tetration in finally a valid field of mathematics research in > its own right, but that it is an active and vital area of > research. Andrew Robbins > (http://wiki.tetration.org/index.php?) is also doing serious > work on tetration and has his own tetration web site > (http://tetration.itgo.com/). > This site is currently primarily Daniel's work although it > does now have Andrew Robbins' Super-logarithm entry > (http://wiki.tetration.org/index.php?title=Super-logarithm). > > From http://en.wikipedia.org/wiki/Tetration: > "At this time there is no commonly accepted solution to the > general problem of extending tetration to the real or complex > numbers, although it is an active area of research." > > Further on this page PowerTower[10,0.99] is estimated to 9.77 > and PowerTower[10,1.01] to 10.55, while I obtain > > PowerTower[10,0.99] = 9.62145617690982427469 > > and > > PowerTower[10,1.01] = 10.4004278027106549883 > > Best regards > > Ingolf Dahl > ingolf.dahl at telia.com > > > > -----Original Message----- > > From: Ingolf Dahl [mailto:ingolf.dahl at telia.com] > > Sent: den 9 december 2006 12:10 > > To: mathgroup at smc.vnet.net > > Subject: [mg72033] REPOSTING: PowerTower extended to real exponents > > > > I am reposting this (probably lost mail due to some > problems with our > > mailserver), with some small corrections in the definition of > > PowerTower: > > > > This is a continuation of the thread [mg71764] Functional > > decomposition (solving f[f[x]] = g[x] for given g), but I > wanted a new > > title. > > > > The function PowerTower gives x to the power x to the power x ... n > > times, and is in the Mathematica Help (for Power) defined as > > > > PowerTower[(x_)?NumericQ, (n_Integer)?Positive] := > Nest[x^#1 & , x, n > > - 1] > > > > This operation is also known as tetration (good Google > search word), > > iterated exponentials and hyperpowers. > > > > I have extended the definition of PowerTower to cover > values Log[x] > > > 0.55 and real values of n >= -2. (Smaller values of x demand a > > slightly different algorithm, which has not been coded yet.) > > > > Some results from this morning: > > > > e tetrated to pi: > > > > Timing[PowerTower[N[E, 50], N[Pi, 50]]] > > > > {2.4530000000000003*Second, > > 3.7150463906547139171107134916520886913907197951098502328119`4 > > 6.532680501112 > > 68*^10} > > > > pi tetrated to e: > > > > Timing[PowerTower[N[Pi, 50], N[E, 50]]] {2.625000000000001*Second, > > 1.885451906681809677772360465630708697760585825573061664`36.29 > > 506406979924*^ > > 6} > > > > 10 tetrated to 2.390797847503886227: > > > > Timing[PowerTower[10., 2.390797847503886227`18.378542856155452]] > > {0.516*Second, 1.0000000000003806*^100} > > > > 2 tetrated to 0.5: > > > > Timing[PowerTower[SetAccuracy[2, 50], 0.5]] > > {0.031000000000000583*Second, > > 1.458798141958706268157503723664793362303661625817509512716`47 > > .9503312794002 > > 24} > > > > Code is below. > > Best regards > > > > Ingolf Dahl > > ingolf.dahl at telia.com > > (snipped, download code from > http://web.telia.com/~u31815170/Mathematica/) > >