User talk:MathFacts/Tetration Summary

Dubious
$$\log_a \frac{f'_a(x)}{f'_a(0)(\ln a)^{x}} = \sum_{n=1}^{\infty} \frac{f^{(n-1)} (0)}{n!} (B_n(x)-B_n)$$
 * Could someone please provide a proof or references for this? AJRobbins (talk) 19:46, 3 September 2009 (UTC)
 * This is based on Faulhaber's formula.--MathFacts (talk) 22:46, 5 September 2009 (UTC)

formula 2 (lagrange)
There occurs an undefined letter "m" in the iteration-parameter of exp, Gottfried —Preceding unsigned comment added by Druseltal2005 (talk • contribs) 05:42, 14 June 2010 (UTC)
 * Thank you. Corrected.--MathFacts (talk) 00:33, 18 September 2010 (UTC)

couldn't compute the regular sexp-version
I tried your formula for regular iteration method in the following implementation without success. Maybe I've some parsing error...

Taken from:
 * 4. Regular iteration method. It is derived using the technique of regular iteration at the fixed point of the exponential function.

\operatorname{sexp}_a(x)=\lim_{n\to\infty} \log_a^{[n]}\left(\left(1-\left(\ln \left(\frac{W(-\ln a)}{-\ln a}\right)\right)^x\right)\frac{W(-\ln a)}{-\ln a}+\ln \left(\frac{W(-\ln a)}{-\ln a}\right)\exp_a^{[n]}(1)\right) $$
 * The expression $$ \frac{W(-\ln a)}{-\ln a} $$ is the principal fixed point of the function $$a^x$$.
 * The expression $$ \frac{W(-\ln a)}{-\ln a} $$ is the principal fixed point of the function $$a^x$$.

For brevity of the formula I assign the fixpoint-expression to the symbol t, and assign the symbol u for u=ln(t) Then I parse the above sexp-equation in the following way:



\operatorname{sexp}_a(x)=\lim_{n\to\infty} \log_a^{[n]}\left(\left(1-u^x\right)*t + u*\exp_a^{[n]}(1)\right) $$ Implementation in Pari/GP is the following: Then a sequence of approximations for sexp(1), n=1..12^2 is For all other x it seems to approximate always sexp(x)→4.0; for instance x=1.1

What am I missing? Gottfried Helms --Gotti 16:40, 1 October 2010 (UTC) —Preceding unsigned comment added by Druseltal2005 (talk • contribs)


 * This is a screenshot of Mathematica notebook which demonstrates the method: http://static.itmages.ru/i/10/1002/h_1286026077_9fee2a059e.png --MathFacts (talk) 13:28, 2 October 2010 (UTC)

It would make sense to discuss Kneser's Riemann mapping method
Kneser's Riemann transformation method is a theoretically proven method. The method starts with a regular superfunction solution developed from the fixed point for bases > exp(1/e), and transforms this entire solution to the real valued sexp at the real axis. The downside, is the difficulty in computing a Riemann mapping. Previously, Jay Daniels Fox had done some medium level precision calculations of a Riemann mapping. I have published a pari-gp program on math.eretrandre.org, entitled kneser.gp which calculates iteratively calculates the Riemann mapping over a wide range of bases, by reinterpreting the Riemann mapping as a mathematically equivalent theta(z) mapping. There are many comments on the website, but I have not yet published a paper. Sheldonison (talk) 11:39, 31 May 2011 (UTC) Sheldonison