sheldonison Wrote:Thank you for your comments. I generated my approximations before seeing "http://math.eretrandre.org/tetrationforum/archive/index.php/thread-33.html" or anything else on this forum. Given that these are the very first posts I have ever made to a math forum, and given that its 30 years since I took calculus, I think I did ok!

Very ok.

(Also a warm somewhat delayed welcome on the forum to you.)

The mentioned article does not disturb your article. There are several possibilities for tetration discussed on this board. JayD has even developed a different tetration approach, which he mentioned in his last post in the thread on sci.math.research. The slog of Andrew is another approach to tetration. There are also the tetration of Dmitrii Kouznetsov and the matrix operator method from Gottfried, as well as the approach of regular iteration for bases (the given names may though not indicate the first inventor of the corresponding method). We still dont know about equality or inequality of these approaches.

So I just wanted to compare your approach with an approach the we not even pursued yet on the forum. Your approach surely can be extended to an arbitrary degree of the polynomials, i.e. to make a series development at . I hope that the corresponding polynomial equation systems can be numerically solved with some math software, and that it has only one real solution. I guess one would come to a different polynomial equation system with a different solution when directly developing a power series at , i.e. without the condition but an arbitrary point . Which is mentioned here.

The critical section is interesting because it is where the point of inflection in the tetration curve is. The critical section is almost linear, and the inflection points position is the most visible feature of this section of curve. That's what motivated the boundary condition for f'(x) = f'(x-1); also it makes the equations simpler. Secondly, the inflection point seems like a universal section of the curve, that applies to any tetration base larger than e^(1/e). The approximation was verified for base 2, e, 3, and base 10.

Quote:The inflection point still seems like an important feature of the tetration curve. Jay's graphs (in the link above), show the inflection points, which is the minimum of the odd derivatives, gradually moving from the ~= -0.5 range to the +0.35 range for higher even derivatives. It would be interesting to see a table of the "x" coordinate of the inflection points, along with the slope of the tetration curve at the point of inflection for various even derivatives.

In that article Jay uses Andrew's slog. The inflection point may vary among the different tetrations.