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 Mizugadro, pentation, Book Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 02/10/2015, 12:07 AM Thank you, MphLee, for your interest. (02/09/2015, 08:19 PM)MphLee Wrote: .. I wonder if you were able to explain to me some of your last results..Yes, I shall try. Quote:(1) for example how do you achieve uniqueness in easy words? I mean, what are the properties that the solution must satisfy and that allow us to derive/(imply) the uniqueness? I postulate the holomorphism in so wide range as I can. I postulate the specific behaviour at infinity. These postulates allow the efficient evaluation. Quote:(2) what is/are your algorithm/s for tetration? For the second question I did my best but I was never able to go through the overwhelming amount of formulas and find something.. Looking at Mizugadro's page of Tetration it says that three different algorithms are used for bases belonging to three different domains [A]$b\in ]1;e^{1/e}[$, [B]$b=e^{1/e}$ and [C]$b\in ]e^{1/e},\infty[$ (let's skip the complex cases) Yes. Let us consider these three cases one by one. Quote:[A] For this case Mizugadro says that the related tetration function ${\rm tet}_b$( aka $1$-based supefunction) is constructed with regular iteration at smallest of the real fixed points $L$ of the function $\log_b$, even if I don't know much about how the regular iteration method works, it says that the function should be of the form $\displaystyle F(z) = L+\sum_{n=1}^{N} a_n {e^{kzn}} + o({e^{kxN}})$ ...but I'm lost before finding the values for $k$.. Let us find you. What happens, it you substitute the representation above into the transfer equation $F(z+1)=\exp_b(F(z))$ ? Please, express the left hand side and the right hand side as series with respect to the small parameter $\varepsilon=\exp(kz)$ and collect terms with the same power of $\varepsilon$. You may compare the result with general formulas (6.3) and (6.4) of my book "Суперфункции" for the specific transfer function $T=\exp_b$ Quote:and for the coefficients $a_n$ in the case of tetration... references of Mizugadro sends me to "D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756. " but the link is broken Please, type the source that indicates the wrong link and indicate, which ULR does not work. I shall try to fix it. Then we'll consider the other your questions. Sincerely. Dmitrii. « Next Oldest | Next Newest »