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 Change of base formula for Tetration sheldonison Long Time Fellow Posts: 622 Threads: 22 Joined: Oct 2008 05/05/2009, 11:21 AM (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 FAQI 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. $\text{sexp}_b(x) = \text{ } \lim_{n \to \infty} \text{log}_b^{\circ n}(\text{sexp}_a (x + \text{slog}_a(\text{sexp}_b(n)))$ 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/tetrationforu...hp?tid=236. tommy1729 Ultimate Fellow Posts: 1,354 Threads: 328 Joined: Feb 2009 05/05/2009, 12:19 PM (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 FAQI 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. $\text{sexp}_b(x) = \text{ } \lim_{n \to \infty} \text{log}_b^{\circ n}(\text{sexp}_a (x + \text{slog}_a(\text{sexp}_b(n)))$ 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/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 sheldonison Long Time Fellow Posts: 622 Threads: 22 Joined: Oct 2008 05/05/2009, 01:29 PM (This post was last modified: 05/05/2009, 03:04 PM by sheldonison.) (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 $\theta(x)$ 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/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, $\theta(\text{slog}_a(x))$. 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. $\text{slog}_2(x) - \text{slog}_e(x) = 1.1282$ $\text{slog}_3(x) - \text{slog}_e(x) = -0.1926$ $\text{slog}_{10}(x) - \text{slog}_e(x) = -1.1364$ bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 05/05/2009, 01:37 PM (This post was last modified: 05/05/2009, 01:40 PM by bo198214.) (05/05/2009, 01:29 PM)sheldonison Wrote: the base conversion wobble term, Do we make this the official name? « Next Oldest | Next Newest »

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