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Change of base formula for Tetration
#41
(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.


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.


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.
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#42
(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.


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.


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
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#43
(05/03/2009, 10:20 PM)tommy1729 Wrote: 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 )
Well, its not that bad, since n is an integer, sexp_b(n) is well defined. You can leave off the 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 where , whose sexp solution I was able to derive, see http://math.eretrandre.org/tetrationforu...236&page=1.

(05/03/2009, 10:20 PM)tommy1729 Wrote: furthermore , i asked how change of base formula for tetration and exp(z) - 1 relate ?

that isnt answered ...
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.

(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, . Here are some examples of base conversion values I derived using sexp derived from base , which has a wobble when compared to Andy's solution or Dimitrii's solution.



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#44
(05/05/2009, 01:29 PM)sheldonison Wrote: the base conversion wobble term,

Do we make this the official name? Big Grin
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