11/17/2014, 09:50 PM
Well hello everybody. It's been a while since I posted here. I've been working vivaciously on iteration and fractional calculus and the ways the two intertwine and I've found a nice fact about tetration.
I've been able to prove an analytic continuation of tetration for bases \( 1 < \alpha < e^{1/e} \) and I've been wondering about the base change formula, if this admits a solution for \( \alpha > e^{1/e} \). The solution I've generated is periodic with period \( 2\pi i / \log(\beta) \) where \( \beta \) is the attracting fixed point of \( 1 < \alpha < e^{1/e} \). The solution is culminated in two papers, where the first paper is purely fractional calculus and the second paper reduces problems in iteration to problems in fractional calculus.
On the whole, I am able to iterate entire functions \( \phi \) with fixed points \( \xi_0 \) such that \( 0<\phi'(\xi_0) < 1 \). And then only able to iterate the function within the region where \( \xi \) is such that \( \lim_{n \to \infty}\phi^{\circ n}(\xi) = \xi_0 \). For exponentiation this implies we can only iterate exponential functions with a fixed point such that its derivative is attracting and positive. This equates to functions with fixed points whose derivative is positive and less than one.
From this I am wondering if it is possible to extend tetration to bases bigger than \( \eta \). Since I have been able to generate tetration bases less than \( \eta \) which are unique and determined by a single equation involving only naltural exponentiations of the number, \( \alpha,\, \alpha^\alpha,\,\alpha^{\alpha^{\alpha}}, \,^4 \alpha,\,.... \). As in we only need know the natural iterates of \( \alpha \) exponentiated, to produce the complex iterates.
I am mostly just wondering about the basechange formula to see if I can generate bases greater than \( e^{1/e} \) using the method I've found. The benefit is that my method is unique, no other function is exponentially bounded which interpolates the values of tetration on the naturals.
All I ask is if anyone can explain the base change formula clearly and if knowing tetration for bases < \eta can extend tetration for greater values.
I have not written my formula for tetration as I am trying to write a paper which contains it in a simple proof. I wish to hide the simple proof ^_^.
I've been able to prove an analytic continuation of tetration for bases \( 1 < \alpha < e^{1/e} \) and I've been wondering about the base change formula, if this admits a solution for \( \alpha > e^{1/e} \). The solution I've generated is periodic with period \( 2\pi i / \log(\beta) \) where \( \beta \) is the attracting fixed point of \( 1 < \alpha < e^{1/e} \). The solution is culminated in two papers, where the first paper is purely fractional calculus and the second paper reduces problems in iteration to problems in fractional calculus.
On the whole, I am able to iterate entire functions \( \phi \) with fixed points \( \xi_0 \) such that \( 0<\phi'(\xi_0) < 1 \). And then only able to iterate the function within the region where \( \xi \) is such that \( \lim_{n \to \infty}\phi^{\circ n}(\xi) = \xi_0 \). For exponentiation this implies we can only iterate exponential functions with a fixed point such that its derivative is attracting and positive. This equates to functions with fixed points whose derivative is positive and less than one.
From this I am wondering if it is possible to extend tetration to bases bigger than \( \eta \). Since I have been able to generate tetration bases less than \( \eta \) which are unique and determined by a single equation involving only naltural exponentiations of the number, \( \alpha,\, \alpha^\alpha,\,\alpha^{\alpha^{\alpha}}, \,^4 \alpha,\,.... \). As in we only need know the natural iterates of \( \alpha \) exponentiated, to produce the complex iterates.
I am mostly just wondering about the basechange formula to see if I can generate bases greater than \( e^{1/e} \) using the method I've found. The benefit is that my method is unique, no other function is exponentially bounded which interpolates the values of tetration on the naturals.
All I ask is if anyone can explain the base change formula clearly and if knowing tetration for bases < \eta can extend tetration for greater values.
I have not written my formula for tetration as I am trying to write a paper which contains it in a simple proof. I wish to hide the simple proof ^_^.