02/20/2009, 02:51 PM
(This post was last modified: 02/20/2009, 05:15 PM by sheldonison.)
bo198214 Wrote:But this question is different from whether \( \text{slog}_a(x)-\text{slog}_b(x) \) converges to a constant, if we assume \( a,b>e^{1/e} \). If \( \text{slog}_a(x)-\text{slog}_b(x) \) does converge for \( x\to\infty \) then - as \( \text{sexp}_b \) goes to infinity - also \( \text{slog}_a(\text{sexp}_b(x))-x \) converges. But this means that \( \text{slog}_a(\text{sexp}_b(x)) \) can not converge to a constant and vice versa.
typo: \( \text{slog}_a(\text{sexp}_b(x))-x \) is what converges to a cyclic 1-cycle function as x grows larger and may converge to a constant as x grows larger for some definitions of slog/sexp.
Quote:But it never gets completely linear, doesnt it? Otherwise it would not be analytic.no, of course not. But I think the contributions of the higher order terms versus a linear estimate between the second and third terms, or between \( \text{log}_b(e) \) and \( \text{log}_b(\text{log}_b(e)) \), becomes insignificant. I plan to try and show that the second and higher order derivatives for the two points, contribute an insignificant delta as b approaches \( e^{1/e} \), and that in the limit, the linear term dominates the higher order terms by an arbitrarily large amount, and that the linear approximation suffices as a definition for the tetration for base b.
I had originally intended this thread to be a search for other links about about base changes for tetration. I seem to have gotten side tracked on to defining the sexp function for real numbers for bases \( >e^{1/e} \).
- Sheldon Levenstein
