(05/05/2009, 11:21 AM)sheldonison Wrote: (05/03/2009, 10:20 PM)tommy1729 Wrote: ???
how does change of base formula for tetration and exp(z) -1 relate ???
i would recommend -- in case this is important and proved -- that more attention is given to it on the forum and/or FAQ
I agree that the base conversion problem is very interesting, and needs more attention.
Here's another way to write the change of base equation, converting from base a to base b.
 = <br />
\text{ } \lim_{n \to \infty} <br />
\text{log}_b^{\circ n}(\text{sexp}_a (x + \text{slog}_a(\text{sexp}_b(n))))
In this equation,
converges to n plus the base conversion constant. This base conversion will have a small 1-cyclic periodic wobble,
, when compared to Dimitrii's solution.
) = <br />
\text{ } \lim_{n \to \infty} <br />
\text{log}_b^{\circ n}(\text{sexp}_a (x + \text{slog}_a(\text{sexp}_b(n))))
Convergence for real values of x is easy to show, and emperically the derivatives appear continuous, but behavior for complex values is a more difficult problem. I would also like to characterize the
sinusoid, and find out whether or not it is c-oo, and whether the sexp_b(z) shows the singularities in the complex plane predicted by Dimitrii Kouznetsov. My thoughts started before I read Jay's post, but you can see them here, http://math.eretrandre.org/tetrationforu...hp?tid=236.
the problem is the wobble ...
its a bit of an illusionary use :
first you give a formula to compute sexp_b(x) bye using sexp_b(n)
then you correct sexp_b(x) to sexp_b(x + wobble(x))
which basicly just means ;
you got a formula for sexp_b(n) using sexp_b(n) ... ?!?
thats pretty lame selfreference ...
( godel escher and bach anyone ? :p )
furthermore , i asked how change of base formula for tetration and exp(z) - 1 relate ?
that isnt answered ...
furthermore i had the idea that
slog_a(x) - slog_b(x) =/= 0 for a =/= b =/= x and a,b,x > e^e
and
slog_a(x)' - slog_b(x)' =/= 0 for a =/= b =/= x and a,b,x > e^e
( derivative with respect to x )
and that this might require a different Coo slog but would imply a uniqueness condition ?
also the equation lim slog_a(oo) - slog_b(oo) = x
seems intresting.
regards
tommy1729