(01/12/2015, 12:06 AM)tommy1729 Wrote: I have trouble with all links that start with " mizugadro " . ..

Perhaps, I should load the most important articles to

http://math.eretrandre.org/hyperops_wiki
It seems to be more stable.

Quote:But I feel the lack of complete explaination and motivation , at least in what I read -at first sight -.

.. Does it has a closed form? How was it computed? Do you mean 1.23515... ? What is the property of this number ; does the iteration only work if 1.23515 log(d) < f(z) < 1.23516 log(d) ?

How did you find that solution ?

I am not sure if I understand you well.. Perhaps, you refer to the Sheldon base..

Sheldon Levenstein asked me to make tetration namely for that base. I describe, how do I find the solution.

First, I implemented function, that evaluates the appropriate fixed points of the exponent and logarithm to the given base, I call it "filog",

http://mizugadro.mydns.jp/t/index.php/Filog
Then, for base \( b \), I POSTULATE, that the \( \mathrm{tet}_b(z) \) should approach

\( \mathrm{filog}(b) \) at \( z\rightarrow \mathrm i \infty \) and

\( \mathrm{filog}(b^*)^* \) at \( z\rightarrow - \mathrm i \infty \)

Then, with complex double arithmetics, the Cauchi integral gives of order of 14 correct decimal digits of \( \mathrm{tet}_b(z) \); I invite you to do it for any base you like.

I did not check that the algorithm converges for every complex base,

but the claim by Sheldon Levenstein, that it fails for the "Sheldon base", is refuted.

Quote:Does the iterate then work for all initial values of z , or do we have a fractal zone where it works ?

Henryk Trappmann asked me to extend the algorithm beyond the cutline, to see the behaviour of another branch.. There is some fractal-like behaviour there. It is not difficult, you also can plot the beautiful pictures.

The algorithm for \( n \)th iterates of a transfer function \( T \) is based on the superfunction \( F \) and abelfuntion \( G \):

\( T^n(z)=F(n+G(z)) \)

Usually, abelfunction \( G \) has branchpoints and cutlines.

Superfunction \( G \) also may have some. So, in some cases, for some regions of argument \( z \), the non-integer iterates may look scratched.

Is this that you ask about?

Quote: Are you talking about bases smaller then eta only in your formula ?

I talk about complex bases. For complex base, in general, the tetration can be evaluated through the Cauchi integral.

Also, it works for real base, greater than eta.

For real base equal or smaller than eta, the algorithm above cannot be applied. For these cases, two other algorithms are suggested, regular iteration and exotic iteration.

I think, all these algorithms evaluate the same tetration, I mean holomorphism of \( \mathrm{tet})b(z) \) with respect to base \( b \) at fixed \( z \).

Quote: Cant we just rewrite 1.23515 by 1.2 and get the same result ?

What does it mean, "the same"?

If the base changes, the explicit plots and the complex maps change. But the result is the same in the sense, that the algorithm allow the evaluation and this evaluation passes series of various tests. We have Henrik Trappmann, who can write the long complicated formulas as a proof, that my simple algorithm provide the unique solution:

http://www.springerlink.com/content/u7327836m2850246/ H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011)

Quote:I prefer color wheel graphs ...

You can plot the graphs in any way you like.

If you meet any difficulties with the running of my algorithms,

or any problems with downloading them from Mizugadro, let me know.

Quote:I hope my criticism does not upset you.

Your criticism is welcomed. I try to answer all your questions.

Quote: For pentation see the comment above + Why the focus on L and exp(kz) ?

Because it provides the efficient way of evaluation of superfunction;

it is not specific for pentation: many other superfunctions can be constructed in this way. For example, superfactorial.

Quote:Tania seems like a sister of Lambert.

Apart from the physics and the identity in the table , does Tania serve another purpose?

Yes. But the Lambert, as it is implemented in Mathematica and other software, has ugly behaviour in the complex plane, while Tania has only two branchpoints and only two cutlines. In order to avoid the confusions, I need the specific name, different from Lambert.

This Tania has many uses.

Quote: I do not know much about Tania functions or the related physics , but I like the idea of naming functions after girls ??

Past century, Tania Kuznetsova asked me to solve numerically some equations, similar to \( f'(x)=f(x)/(1+f(x)) \)

Yes, it comes from physics. We have published the serious of articles.. May be, one day, I'll describe how did that happen, it may be as a funny detective story.

Quote:MAYBE you said this before but why did you choose - assuming choice !? - pentation to be periodic ?

I think, I am first to provide the efficient algorithm for evaluation of superfunction of tetration, and to provide the detailed complex map of this superfunction, and to publish this. I use my prerogative to choose name for this function. I call it <b>pentation</b> and denote with symbol pen. By construction, this function is periodic. The period is determined by the derivative of tetration at its "zeroth" fixed point.

You may chose the specific base, \( b=\tau\approx 1.63532 \), such that graphic \( y=\mathrm{tet}_{\tau}(x) \) touches the graphic of the identity function, \( y=x \), and for this base construct the non-periodic supertetration.

The method is described in

http://www.ils.uec.ac.jp/~dima/PAPERS/2011e1e.pdf
http://mizugadro.mydns.jp/PAPERS/2011e1e.pdf H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of computation, 2012 February 8. ISSN 1088-6842(e) ISSN 0025-5718(p)

I suggest, that you call this supertetration with some specific name, for example, SuTet (Super Tetration), in order to keep name pen (pentation) for the periodic superfunction of tetration, id est, the superfunction, shown in the map above.

I think, you can implement the algorithm and plot the pictures and describe them by yourself, and submit them to some "Mathematics of Computation". If finish it within few months, I shall be glad to cite your publication in the English version of my Book. If you need any kind of help about this, let me know.