Mizugadro, pentation, Book

[Kouznetsov]

..

http://mizugadro.mydns.jp/t/index.php/Tetration and looked at the entry for the tetration to base b=exp(1/e). I do not understand how you arrive at the polynomials, and especially how at the set of power-series in the table with the increasing m.

..
Well, here I not even know what you might mean with "primitive fit". Can I find this in your Tori-wiki?

Yes.
http://mizugadro.mydns.jp/t/index.php/Fit1.cin

This file is cited at http://mizugadro.mydns.jp/t/index.php/Fi...0bx10d.png
which is cited at
http://mizugadro.mydns.jp/t/index.php/Tetration

Today, I had some crash at Mizugadro. I have recovered it, but I did not find its reason. For the case if any similar happen again, there is copy at
http://en.citizendium.org/wiki/File:Tetreal10bx10d.png

If I find it I'd try this using Pari/GP to see myself, trying to use greater precision, whether my approach can be finetuned to arrive at your values at all.

I think, first we should compare our results for some fixed value of base,
for example, for natural tetration , or for b=sqrt(2).
Then we may consider the derivatives with respect to base.

.. Can you evaluate tetration to the complex base?
.. Well, this would only be the next step, I'm afraid. If I cannot reproduce your values in the simple (=real) case, why invest more work to implement the even more difficult complex bases...

My approach implies complex base and complex argument.
You deal with the primitive fit, instead of to compare with advanced approximations, that are declared to provide 14 digits.

If you want to deal with the Taylor expansion,
you may begin with the Taylor expansion of natural tetration at zero.
It is shown in figure 14.6 of my book and also loaded to TORI at
http://mizugadro.mydns.jp/t/index.php/File:Vladi04.jpg
http://mizugadro.mydns.jp/t/index.php/Vladinaiv49.cin

Ah, p.s.: instead of the word "preliminary" in the exp(1/e)-section - did you possibly mean "provisorical"?

Many times I saw word "preliminary" in the scientific texts, and I think, I know its meaning. But I do not know the meaning of word "provisorical". So, type "preliminary".
I think, in the way I had mentioned, we may get of order of 14 digits for some tens by tens coefficients. However, first, tetration of the complex argument and complex base should be evaluated with similar precision.

Fist, efficient algorithms of the evaluation and the tests, verifications.
Then, any expansion you like.