11/05/2009, 03:00 AM
(This post was last modified: 11/05/2009, 09:14 PM by dantheman163.)
I believe I have found an analytic extension of tetration for bases 1 < b <= e^(1/e).
This is based on the assumption
(1) The function y=b^^x is a smooth, monotonic concave down function
Conjecture:
If assumption (1) is true then
\( {}^x b = \lim_{k\to \infty} (log_{b}^{ok}(x({}^k b- {}^{(k-1)} b)+{}^k b) ) \) for \( -1 \le x\le 0 \)
Some properties:
This formula converges rapidly for values of b that are closer one.
For base eta it converges to b^^x for all x but this is not true for the other bases.
Interestingly for b= sqrt(2) and x=1 it seems to be converging to the super square root of 2
I will try to post a proof in the next couple of days I just need some time to type it up.
Thanks
This is based on the assumption
(1) The function y=b^^x is a smooth, monotonic concave down function
Conjecture:
If assumption (1) is true then
\( {}^x b = \lim_{k\to \infty} (log_{b}^{ok}(x({}^k b- {}^{(k-1)} b)+{}^k b) ) \) for \( -1 \le x\le 0 \)
Some properties:
This formula converges rapidly for values of b that are closer one.
For base eta it converges to b^^x for all x but this is not true for the other bases.
Interestingly for b= sqrt(2) and x=1 it seems to be converging to the super square root of 2
I will try to post a proof in the next couple of days I just need some time to type it up.
Thanks
