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+--- Thread: Change of base formula for Tetration (/showthread.php?tid=14)
RE: Change of base formula for Tetration - sheldonison - 05/05/2009
(05/03/2009, 10:20 PM)tommy1729 Wrote: ???I agree that the base conversion problem is very interesting, and needs more attention.
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
Here's another way to write the change of base equation, converting from base a to base b.
\( \text{sexp}_b(x) =
\text{ } \lim_{n \to \infty}
\text{log}_b^{\circ n}(\text{sexp}_a (x + \text{slog}_a(\text{sexp}_b(n))) \)
In this equation, \( \text{slog}_a(\text{sexp}_b(n))) \) converges to n plus the base conversion constant. This base conversion will have a small 1-cyclic periodic wobble, \( \theta(x) \), when compared to Dimitrii's solution.
\( \text{sexp}_b(x+\theta(x)) =
\text{ } \lim_{n \to \infty}
\text{log}_b^{\circ n}(\text{sexp}_a (x + \text{slog}_a(\text{sexp}_b(n))) \)
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 \( \theta(x) \) 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/tetrationforum/showthread.php?tid=236.
RE: Change of base formula for Tetration - tommy1729 - 05/05/2009
(05/05/2009, 11:21 AM)sheldonison Wrote:(05/03/2009, 10:20 PM)tommy1729 Wrote: ???I agree that the base conversion problem is very interesting, and needs more attention.
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
Here's another way to write the change of base equation, converting from base a to base b.
\( \text{sexp}_b(x) =
\text{ } \lim_{n \to \infty}
\text{log}_b^{\circ n}(\text{sexp}_a (x + \text{slog}_a(\text{sexp}_b(n))) \)
In this equation, \( \text{slog}_a(\text{sexp}_b(n))) \) converges to n plus the base conversion constant. This base conversion will have a small 1-cyclic periodic wobble, \( \theta(x) \), when compared to Dimitrii's solution.
\( \text{sexp}_b(x+\theta(x)) =
\text{ } \lim_{n \to \infty}
\text{log}_b^{\circ n}(\text{sexp}_a (x + \text{slog}_a(\text{sexp}_b(n))) \)
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 \( \theta(x) \) 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/tetrationforum/showthread.php?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
RE: Change of base formula for Tetration - sheldonison - 05/05/2009
(05/03/2009, 10:20 PM)tommy1729 Wrote: the problem is the wobble ...Well, its not that bad, since n is an integer, sexp_b(n) is well defined. You can leave off the \( \theta(x) \) function, you just get a different solution, one that wobbles a little bit, easier to see in the higher derivatives. Also, in my original post, I was using a home base of \( \eta+\delta \) where \( \eta=e^{1/e} \), whose sexp solution I was able to derive, see http://math.eretrandre.org/tetrationforum/showthread.php?tid=236&page=1.
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 )
(05/03/2009, 10:20 PM)tommy1729 Wrote: furthermore , i asked how change of base formula for tetration and exp(z) - 1 relate ?Jay isn't around to answer. He discusses base change convergence, which I understand perfectly well. But I didn't understand the double logarithmic paragraph. Jay abandoned this approach to tetration, because it gives different results than Andrew Robbin's solution, (and Dimitrii Kouznetsov's solution) due to the wobble.
that isnt answered ...
(05/03/2009, 10:20 PM)tommy1729 Wrote: furthermore i had the idea that
slog_a(x) - slog_b(x) =/= 0 for a =/= b =/= x and a,b,x > e^e
For large enough values of x, slog_a(x) - slog_b(x) will converge to a specific value. That value will be the sexp base conversion constant plus the base conversion wobble term, \( \theta(\text{slog}_a(x)) \). Here are some examples of base conversion values I derived using sexp derived from base \( \eta^{+} \), which has a wobble when compared to Andy's solution or Dimitrii's solution.
\( \text{slog}_2(x) - \text{slog}_e(x) = 1.1282 \)
\( \text{slog}_3(x) - \text{slog}_e(x) = -0.1926 \)
\( \text{slog}_{10}(x) - \text{slog}_e(x) = -1.1364 \)
RE: Change of base formula for Tetration - bo198214 - 05/05/2009
(05/05/2009, 01:29 PM)sheldonison Wrote: the base conversion wobble term,
Do we make this the official name?
