04/14/2015, 04:48 PM
(This post was last modified: 04/15/2015, 03:23 PM by sheldonison.)
(04/14/2015, 02:36 PM)sheldonison Wrote: \( L\approx0.277987425;\;\;\; \lambda\approx -1.28017940 \)
...
Normally, we would use \( f(z)=S(\lambda^z) \) for the superfunction, but since lambda is a negative number, perhaps one could use \( f(z)=S\left( (-\lambda)^z \cdot \sin(\pi z)\right) \) to generate the exponentially increasing pseudo 2-periodic solution the Op is looking for.
\( S(z) = \sum_{n = 0}^{\infty} a_n \cdot z^n\;\;\; S(\lambda z) = {0.01}^{S(z)}\;\;\; S(0)=L \;\;\; {0.01}^L=L \;\;\; \lambda \approx -1.28017940081259 \)
Code:
a0= 0.277987424809561
a1= 1
a2= 1.00982628479736
a3= -1.74658670929184
a4= -3.98120395207580
a5= 3.59453988520567
a6= 13.8485552663038
a7= -6.48959552733049
a8= -44.4747907113725
a9= 7.91461618658992And here is a graph, showing an analytic super-function for \( {0.01}^z \) the Op might be interested in, from -10 to 10, where it starts out oscillating around the primary fixed point, and then converges to the two cycle that the Op noted. Of course, this isn't tetration; Tet(-2) is by convention a logarithmic singularity. And this function has no uniqueness, I could have just as easily used cosine instead of sine, or any of other 2-cyclic function with \( \theta(z+1)=-\theta(z) \).
- Sheldon
