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REPOSTING:RE: REPOSTING: PowerTower extended to real exponents

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  • Subject: [mg72200] REPOSTING:RE: [mg72033] REPOSTING: PowerTower extended to real exponents
  • From: "Ingolf Dahl" <ingolf.dahl at>
  • Date: Thu, 14 Dec 2006 05:49:22 -0500 (EST)

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.

"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."

"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

"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


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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