03/09/2009, 06:34 PM
(This post was last modified: 03/10/2009, 03:51 PM by sheldonison.)
\( \theta(x) = \text{slog}_{v2}(\text{sexp}_{v1}(x))-x \)
Back to characterizing the \( \theta \) for the published sexp base e equation (version v1), compared to base 1.45 approximated with a 3rd order slog (version v2). At one point I thought the theta might be a relatively simple sin(2*pi*x+phase) type of function, but there are definitely higher harmonics, although the second harmonic sin(4*pi*x) is about 120 times smaller than the main harmonic.
Given everything I know so far, the proposal is the following limit, using a linear approximation for the critical section for slog/sexp b
\( \text{sexp}_c(x) =
\lim_{b \to \eta^+}\text{ } \lim_{n \to \infty}
\text{log}_c^{\circ n}(\text{sexp}_b (x + \text{slog}_b(\text{sexp}_c(n))) \)
converges, and all of its derivatives are continuous. For each base=c, there's a 1-cyclic \( \theta \) function, linking this sexp definition with the traditional sexp function. Unless \( \theta \) is badly behaved, the \( \text{sexp}_{v2}(x)=\text{sexp}_{v1}(x+\theta(x)) \) will also be continuous for all of its derivatives, and possibly analytic. This isn't new, Dimitrii points out on his wiki that the theta transfer function allows constructing other solutions of tetration with a reduced range of convergence in the complex plane, due to singularities caused by theta. The only advantage of this particular alternate sexp definition is that it has constant base conversions between any two bases. But the addition of the theta function introduces a wobble, that becomes more apparent in the higher derivatives; the odd derivatives are no longer positive for all x>-2. Finally, for smaller values of bases, the magnitude of the 1-cyclic \( \theta \) function, generated by comparing the two sexp functions for the same base, gets arbitrarily small as the base=c approaches e^(1/e). This is also critical to the convergence of the limit equation.
Back to characterizing the \( \theta \) for the published sexp base e equation (version v1), compared to base 1.45 approximated with a 3rd order slog (version v2). At one point I thought the theta might be a relatively simple sin(2*pi*x+phase) type of function, but there are definitely higher harmonics, although the second harmonic sin(4*pi*x) is about 120 times smaller than the main harmonic.
Given everything I know so far, the proposal is the following limit, using a linear approximation for the critical section for slog/sexp b
\( \text{sexp}_c(x) =
\lim_{b \to \eta^+}\text{ } \lim_{n \to \infty}
\text{log}_c^{\circ n}(\text{sexp}_b (x + \text{slog}_b(\text{sexp}_c(n))) \)
converges, and all of its derivatives are continuous. For each base=c, there's a 1-cyclic \( \theta \) function, linking this sexp definition with the traditional sexp function. Unless \( \theta \) is badly behaved, the \( \text{sexp}_{v2}(x)=\text{sexp}_{v1}(x+\theta(x)) \) will also be continuous for all of its derivatives, and possibly analytic. This isn't new, Dimitrii points out on his wiki that the theta transfer function allows constructing other solutions of tetration with a reduced range of convergence in the complex plane, due to singularities caused by theta. The only advantage of this particular alternate sexp definition is that it has constant base conversions between any two bases. But the addition of the theta function introduces a wobble, that becomes more apparent in the higher derivatives; the odd derivatives are no longer positive for all x>-2. Finally, for smaller values of bases, the magnitude of the 1-cyclic \( \theta \) function, generated by comparing the two sexp functions for the same base, gets arbitrarily small as the base=c approaches e^(1/e). This is also critical to the convergence of the limit equation.
