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RE: RE: REPOSTING: PowerTower extended to real exponents

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

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

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

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 I have just found it
myself (by a tip in a post in
%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

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] 
> Sent: den 14 december 2006 11:49
> To: mathgroup at
> 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 
> 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
> "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
> "Extending Tetration
> Many people are interested in extending tetration from the 
> natural numbers to the real and complex numbers. Ioannis Galidakis
> ( 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
> ( is also doing serious 
> work on tetration and has his own tetration web site 
> (
> This site is currently primarily Daniel's work although it 
> does now have Andrew Robbins' Super-logarithm entry 
> ( 
> From
> "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
> > -----Original Message-----
> > From: Ingolf Dahl [mailto:ingolf.dahl at]
> > Sent: den 9 december 2006 12:10
> > To: mathgroup at
> > 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
> (snipped, download code from 

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