01/27/2015, 12:28 AM
(This post was last modified: 01/27/2015, 03:06 PM by sheldonison.)

where k is a small perturbation constant in the neighborhood of zero, what if we develop the superfunction of

where b is the corresponding tetration base to k

I plan on developing these ideas over the next several weeks or months ... feel free to contribute or critique. In particular, for a value of k, can we develop the Fatou coordinate/Abel function, and the superfunction? Perhaps involving quasi conformal mappings and parabolic implosion, and perturbed fatou coordinates, which unfortunately, I don't understand that much. The math is very hard, and some of the best research is in French.

k=0 corresponds to eta, which is the singularity point since zero is z fixed point. The idea is that each positive value of k corresponds directly to tetration for some real base greater than eta. One idea is that also, there is some similar function to f(x), which has the same x^2 multiplier, and which has a well defined superfunction which can be expressed in a closed form in terms of tan(bx); see previous mathstack question. The goal would be to develop an Abel/Fatou function for arbitrary values of k...

Here is a graph of the superexponential corresponding to k=0.1, from -5 to 7. The inflection point is always about halfway between 0 and 1, and the superexponential is developed at z=0. k=0.1 corresponds to tetration b=1.501657. Where f(0)=0, this corresponds to where the

Another idea is to develop an asymptotic formal series for the superfunction of f in tems of the pertubation value of k; I posted some preliminary results for such a formal series below. I think both ideas have promise. I see the possibility for a more rigorous definition of complex base tetration coming out of these ideas. So far, I got a pretty good asymptotic (non-converging) series for the first few derivatives of tetration base e, at z=-1. This would be plugging in k=1. For smaller values of k, correspondign to real bases and complex bases closer to eta, the asymptotic series converges better before diverging. Anyway, I'm not thrilled with the series since it is an asymptotic non-converging series...

where b is the corresponding tetration base to k

I plan on developing these ideas over the next several weeks or months ... feel free to contribute or critique. In particular, for a value of k, can we develop the Fatou coordinate/Abel function, and the superfunction? Perhaps involving quasi conformal mappings and parabolic implosion, and perturbed fatou coordinates, which unfortunately, I don't understand that much. The math is very hard, and some of the best research is in French.

k=0 corresponds to eta, which is the singularity point since zero is z fixed point. The idea is that each positive value of k corresponds directly to tetration for some real base greater than eta. One idea is that also, there is some similar function to f(x), which has the same x^2 multiplier, and which has a well defined superfunction which can be expressed in a closed form in terms of tan(bx); see previous mathstack question. The goal would be to develop an Abel/Fatou function for arbitrary values of k...

Here is a graph of the superexponential corresponding to k=0.1, from -5 to 7. The inflection point is always about halfway between 0 and 1, and the superexponential is developed at z=0. k=0.1 corresponds to tetration b=1.501657. Where f(0)=0, this corresponds to where the

Another idea is to develop an asymptotic formal series for the superfunction of f in tems of the pertubation value of k; I posted some preliminary results for such a formal series below. I think both ideas have promise. I see the possibility for a more rigorous definition of complex base tetration coming out of these ideas. So far, I got a pretty good asymptotic (non-converging) series for the first few derivatives of tetration base e, at z=-1. This would be plugging in k=1. For smaller values of k, correspondign to real bases and complex bases closer to eta, the asymptotic series converges better before diverging. Anyway, I'm not thrilled with the series since it is an asymptotic non-converging series...

Code:

`a0= 0`

a1= k + 1/12*k^2 + 1/120*k^3 + 1/5040*k^4 - 1/10080*k^5 + 1/332640*k^6 -323/86486400*k^7

a2= -1/4*k^2 - 1/16*k^3 - 1/90*k^4 - 11/12096*k^5 + 19/1209600*k^6 + 683/79833600*k^7

a3= 1/6*k^2 + 1/8*k^3 + 11/240*k^4 + 1957/181440*k^5 + 5191/3628800*k^6 + 2791/19958400*k^7

a4= -5/48*k^3 - 23/288*k^4 - 121/3456*k^5 - 28303/2903040*k^6 -31687/17418240*k^7

a5= 1/30*k^3 + 13/180*k^4 + 247/4320*k^5 + 39913/1451520*k^6 + 151307/17418240*k^7

a6= -49/1440*k^4 - 469/8640*k^5 - 15053/345600*k^6 - 274711/12441600*k^7

a7= 17/2520*k^4 + 1861/60480*k^5 + 241/5600*k^6 + 1498627/43545600*k^7

a8= - 79/8064*k^5 - 983/35840*k^6 - 204851/5806080*k^7

substituting in k=1, we get a reasonably good approximation for sexp_e at z=-1

a0= 0

a1= 1.09176514457764

a2= -0.324496239678531

a3= 0.349856276054193

a4= -0.230607971873163

a5= 0.198915562077454

a6= -0.153946357381687

a7= 0.114967367541152

a8= -0.0725062348434744

For comparison the "correct" values of the Taylor series coefficients at z=-1 are:

a0= 0

a1= 1.09176735125832

a2= -0.324494761735110

a3= 0.349836269767157

a4= -0.230854426837443

a5= 0.201330212284523

a6= -0.164352165253219

a7= 0.142836335724573

a8= -0.124694993215245

- Sheldon