03/01/2009, 12:18 PM
(This post was last modified: 03/03/2009, 04:44 PM by sheldonison.)
More empirical data on the 3rd order derivatives shows that they're all sinusoidal, and the sinusoidal relative contribution to the third order derivative is seems more or less constant across a large range of sexp base values. This means the 3rd order derivative sinusoid may not change the convergence equations, and may give more direction on the value of the error terms in proving the convergence of:
\( \text{sexp}_e(x) =
\lim_{b \to \eta^+}\text{ } \lim_{n \to \infty} \text{ln(ln(ln(}\cdots
\text{sexp}_b (x + \text{slog}_b(\text{sexp}_e(n)))))) \)
So, here's the graphs, centered on the critical section for several different base conversions. The 3rd derivative is the lowest derivative to show the sinusoid, and it seems to be about 1/6th of the value of the 3rd derivative, and the phase varies. Beginning with the fifth derivative, the sexp extension via base conversion will not have have positive values for odd deriivatives for all x values >-2; so this is clearly a different extension of sexp to reals then Dimittri's extension of sexp to real numbers. Also, the graphs, but for base 1.485 and base 1.6, the results are consistent, converting from base 1.45 and base 1.44533. For converting to base "e", the same 3rd derivative sinusoid is probably there, but it is difficult to see because the 3rd derivative climbs so quickly to infinity on either side of the critical section.
![[Image: base_1447_3rd.gif]](http://www.sheltx.com/share_stuff/base_1447_3rd.gif)
![[Image: base_145_3rd.gif]](http://www.sheltx.com/share_stuff/base_145_3rd.gif)
![[Image: base_1485_3rd.gif]](http://www.sheltx.com/share_stuff/base_1485_3rd.gif)
![[Image: base_16_3rd.gif]](http://www.sheltx.com/share_stuff/base_16_3rd.gif)
\( \text{sexp}_e(x) =
\lim_{b \to \eta^+}\text{ } \lim_{n \to \infty} \text{ln(ln(ln(}\cdots
\text{sexp}_b (x + \text{slog}_b(\text{sexp}_e(n)))))) \)
So, here's the graphs, centered on the critical section for several different base conversions. The 3rd derivative is the lowest derivative to show the sinusoid, and it seems to be about 1/6th of the value of the 3rd derivative, and the phase varies. Beginning with the fifth derivative, the sexp extension via base conversion will not have have positive values for odd deriivatives for all x values >-2; so this is clearly a different extension of sexp to reals then Dimittri's extension of sexp to real numbers. Also, the graphs, but for base 1.485 and base 1.6, the results are consistent, converting from base 1.45 and base 1.44533. For converting to base "e", the same 3rd derivative sinusoid is probably there, but it is difficult to see because the 3rd derivative climbs so quickly to infinity on either side of the critical section.
