04/15/2015, 04:11 AM
(This post was last modified: 04/15/2015, 04:37 AM by sheldonison.)
(04/14/2015, 08:49 PM)marraco Wrote: I conjecture that tetration base 0 should be a discontinuous function, alternating between the values 0 and 1, with period 2, and as the base approach zero, the negative axis turns into a real, continuous function.Maybe; first figure out the bases in between...
Quote:The derivative must converge to a periodic and alternating Diract Delta, multiplied by c₁-c₂, and the surface of each "Diract" must be constant on the positive axis.
That's because there is at least one point in each period with value c₁ and c₂.
Cool stuff! I think the next step is to figure out the Taylor series using some of the techniques we've developed for Tetration. Here, I spent a few minutes generating a four term series approximation using a variant of Andrew's slog technique. This approximation is good between z=-1, and z=0. Taking the \( \ln(\text{tet}(z))\approx \text{tet}(-1+z)\; \); the 2nd derivative is continuous for a piecemeal approximation. The other boundary requirements are: tet(0)=1; tet(-1)=0; tet'(0)=0; tet'(-1)=0;
\( \text{tet_{0.01}(z) \approx 1 -4.92955971 \cdot z^2 - 5.85911942 \cdot z^3 -1.92955971 \cdot z^4 \)
The tet(z) function is real valued for z>-3.9... surprisingly remaining real valued after the logarithmic singularity at -2. I expect there is a unique solution for the complex plane, which can be developed using Kneser's Riemann mapping and the equivalent \( \theta(z) \) mapping techniques we've invented ... which should be really interesting too! We expect this function to approach two of the function's other complex fixed points, and its conjugate, as \( \Im(z) \) gets arbitrarily large, positive or negative.
- Sheldon
