04/16/2015, 05:09 PM
(This post was last modified: 04/16/2015, 09:02 PM by sheldonison.)
Conjecture: For all bases between \( 0<b<1 \), there are a pair of complex conjugate fixed points. Each of these fixed points has a Koenig solution, that can be used to generate a complex valued superfunction. Such a Koenig solution is entire, and never takes on the value of 0. The conjecture is the pair of Koenig complex superfunctions can be Kneser (Riemann) mapped (or equivalently) theta mapped, to a Tetration solution that is real valued between the singularity at -2, and infinity. Such a solution would be like tetration for bases>\( \eta=\exp(1/e) \) analytic if \( \Im(z)>0 \) or \( \Im(z)<0 \) and at the real axis, for z>-2, with singularities at -2, -3, and other values <-3.
For tetration bases \( 0<b<1 \), there is a primary real valued fixed point, with a negative \( \lambda \) multiplier at the fixed point. For bases \( 0<b<\exp(-e) \) the fixed point is repelling, but bifurcates to an attracting two-cycle. marraco and I have posted plausible real valued tetration solutions with Tet(0)=1 for base b=0.01. For bases>\( \exp(-e) \) the fixed point is attracting, and the solution will slowly oscillate towards the fixed point, exponentially scaling as it does so. The initial seed for such a solution would be generated in much the same way as the earlier post for b=0.01; I haven't done the Kneser/theta mapping yet, from the complex pair of fixed points.
For example, consider b=0.1, which has an attracting fixed point \( \approx 0.399 \), and a pair of repelling complex conjugate fixed points \( -0.3018\pm1.981i \). Here is an initial seed, a 4th degree polynomial approximation roughly accurate between tet(-1) and tet(0), with boundary conditions tet(-1)=0, tet(0)=1, tet'(-1)=0, tet'(0)=0, and the \( \ln_b(\text{tet}(x+0))\approx \text{tet}(-1+x) \) with a piecemeal approximation having a continuous 2nd derivative:
\( \text{tet}_{0.1}(z) \approx 1 -4.18324136x^2 -4.36648272*x^3 -1.18324136x^4 \)
Then this is what it looks like, at the real axis from -3.76 to 10. As \( \Re(z) \to \infty, \; \text{tet}(z) \to L \approx 0.399\;\;\; 0.1^{L}=L \)
And here is the function at \( \Im(z)=0.1i \) from -\( -4<\Re(z)<5 \) Notice that the function would go to the secondary fixed point of \( L2 \approx-0.3018-1.981i\;0.1^{L2}=L2 \) as \( \Re(z) \to -\infty \). For \( \Im(z)=-0.1i \), it would go to the conjugate fixed point, \( \approx-0.3018+1.981i \). So this graph is why I think there is a unique Kneser mapping to generate this Tetration function, with the boundary conditions that tet(-1)=0, tet(0)=1, tet'(-1)=0, tet'(0)=0, \( \text{tet}(\Im(z)=\pm i\infty)\approx 0.3018\mp 1.981i\;\; \) (approaching Koenig's solution)
The algorithm just sketched could work for all bases \( 0<b<1 \).
I also have some bonus questions. Is it possible that we derive the requirement that tet'(-1)=0 from the requirement that the Riemann mapping go to the pair of conjugate fixed points in the complex plane? btw, if Tet'(-1)=0, then it is trivial to show that tet'(0)=0, from the definition of the derivative of exp(f(x))
Also, this function is different than Tetration \( \exp(-e) \),http://math.eretrandre.org/tetrationforu...0&pid=6748. Mike and I think that the function at this link is actually the Tetration function that is analytically connected to real base tetration for b>eta. What happens if we start with this new Tetration function, for 0<b<1, and slowly move the base in the complex plane, avoiding the singularities at 1 and eta, back to the real axis for base(e)? Does this new function have a natural analytic boundary someplace, perhaps at the Shell Thron boundary or something like that???? That's a wild guess, btw. For example returning to base(e), I think you wind up with tetration from the secondary fixed point~=2.0623+ 7.589i, in the upper half of the complex plane, along with 0.318-1.337i in the lower half of the complex plane, so this could be challenging... see the somewhat related post tetration from secondary fixed point.
For tetration bases \( 0<b<1 \), there is a primary real valued fixed point, with a negative \( \lambda \) multiplier at the fixed point. For bases \( 0<b<\exp(-e) \) the fixed point is repelling, but bifurcates to an attracting two-cycle. marraco and I have posted plausible real valued tetration solutions with Tet(0)=1 for base b=0.01. For bases>\( \exp(-e) \) the fixed point is attracting, and the solution will slowly oscillate towards the fixed point, exponentially scaling as it does so. The initial seed for such a solution would be generated in much the same way as the earlier post for b=0.01; I haven't done the Kneser/theta mapping yet, from the complex pair of fixed points.
For example, consider b=0.1, which has an attracting fixed point \( \approx 0.399 \), and a pair of repelling complex conjugate fixed points \( -0.3018\pm1.981i \). Here is an initial seed, a 4th degree polynomial approximation roughly accurate between tet(-1) and tet(0), with boundary conditions tet(-1)=0, tet(0)=1, tet'(-1)=0, tet'(0)=0, and the \( \ln_b(\text{tet}(x+0))\approx \text{tet}(-1+x) \) with a piecemeal approximation having a continuous 2nd derivative:
\( \text{tet}_{0.1}(z) \approx 1 -4.18324136x^2 -4.36648272*x^3 -1.18324136x^4 \)
Then this is what it looks like, at the real axis from -3.76 to 10. As \( \Re(z) \to \infty, \; \text{tet}(z) \to L \approx 0.399\;\;\; 0.1^{L}=L \)
And here is the function at \( \Im(z)=0.1i \) from -\( -4<\Re(z)<5 \) Notice that the function would go to the secondary fixed point of \( L2 \approx-0.3018-1.981i\;0.1^{L2}=L2 \) as \( \Re(z) \to -\infty \). For \( \Im(z)=-0.1i \), it would go to the conjugate fixed point, \( \approx-0.3018+1.981i \). So this graph is why I think there is a unique Kneser mapping to generate this Tetration function, with the boundary conditions that tet(-1)=0, tet(0)=1, tet'(-1)=0, tet'(0)=0, \( \text{tet}(\Im(z)=\pm i\infty)\approx 0.3018\mp 1.981i\;\; \) (approaching Koenig's solution)
The algorithm just sketched could work for all bases \( 0<b<1 \).
I also have some bonus questions. Is it possible that we derive the requirement that tet'(-1)=0 from the requirement that the Riemann mapping go to the pair of conjugate fixed points in the complex plane? btw, if Tet'(-1)=0, then it is trivial to show that tet'(0)=0, from the definition of the derivative of exp(f(x))
Also, this function is different than Tetration \( \exp(-e) \),http://math.eretrandre.org/tetrationforu...0&pid=6748. Mike and I think that the function at this link is actually the Tetration function that is analytically connected to real base tetration for b>eta. What happens if we start with this new Tetration function, for 0<b<1, and slowly move the base in the complex plane, avoiding the singularities at 1 and eta, back to the real axis for base(e)? Does this new function have a natural analytic boundary someplace, perhaps at the Shell Thron boundary or something like that???? That's a wild guess, btw. For example returning to base(e), I think you wind up with tetration from the secondary fixed point~=2.0623+ 7.589i, in the upper half of the complex plane, along with 0.318-1.337i in the lower half of the complex plane, so this could be challenging... see the somewhat related post tetration from secondary fixed point.
- Sheldon
