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

for

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

for

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