08/08/2010, 07:14 PM
(This post was last modified: 08/12/2010, 04:36 AM by sheldonison.)
This is a continuation of the thread in the computation forum, http://math.eretrandre.org/tetrationforu...hp?tid=486
This thead contains some of the mathematical equations I used for the fast Kneser algorithm. In this post, B is the base for the sexp function, and L is the fixed point for base B.
This is the complex valued superfunction, developed from the fixed point L for base B, where B>\( \eta \)
\( c=L\times\ln(B) \) for base e, c=L
\( \operatorname{superf}_{B}(z) = \lim_{n \to \infty}
B^{[n](L + c^{z-n})} =
\lim_{n \to \infty} \exp^{[n]} ((L+c^{z-n})/\ln(B)) \)
This is the complex valued inverse superfunction, developed from the fixed point, which is the inverse of the equation above. The inverse superfunction has the property, that isuperf(B^z)=isuperf(B)+1. This particular equation is normalized, so that it converges to the same value as the limit of n approaches infinity. Both of these two functions are implemented in the pari-GP program I wrote.
\( \operatorname{isuperf}_{B}(z) = \lim_{n \to \infty}
\log_{c} (c^n \times ((\log_{B}^{[n]}(z))-L)) \)
If we started with a pefect sexp(z) function, then this is the 1-cyclic theta function linking the sexp with the superf/isuperf.
\( \theta(z)=\operatorname{isuperf}(\operatorname{sexp}(z))-z \)
\( \operatorname{sexp}(z)=\operatorname{superf}(z+\theta(z)) \)
Theta(z) has a singularity at all integer values of n. Theta(z) is represented by an infinite sequence of fourier terms. The fourier series for theta(z) can be developed from any arbitrary unit length on the real axis of sexp(z), where z>-2. Only terms with positive values of n are included, and all terms a_n for negative values of n are zero.
\( \theta(z)=\sum_{n=0}^{\infty}a_n\times \e^{(2\pi n i z)} \)
Theta(z) is intimately connected to the Riemann unit circle mapping, used by Kneser's construction. The Taylor series for the Riemann unit circle function (I'm not sure of the correct notation here) uses the exactly the same a_n coeffecients as the 1-cyclic theta function! This is something that connects the complex fourier analysis of theta(z) to the theory of complex analytic functions, which is really neat! The RiemannCircle has a singularity at z=1, which corresponds to the singularities at the integer values of theta(z).
\( \operatorname{RiemannCircle}(z) = \theta(\ln(z)/2\pi i) \)
\( \theta(z) = \operatorname{RiemannCircle}(e^{2\pi i z}) =\operatorname{isuperf}(\operatorname{sexp}(z))-z \)
\( \operatorname{RiemannCircle(z)=\sum_{n=0}^{\infty}a_n\times z^n \)
If we had a perfect sexp(z) Taylor series, then we have a function for the values of the Riemann unit circle function, which is generated from the theta function, using the equation above, from the inverse superfunction. Now, we can use Cauchy's integral formula to calculate the Taylor series for the Riemann unit circle function. And this also gives us the coefficients of the 1-cyclic theta(z) function.
Of course, there is still the problem of the singularity on the unit circle, which causes problems due to slow convergence. In later posts, I will try to go into some detail, showing values for the Taylor series results for the Riemann circle function, and how the coefficients slowly decay, with poor convergence on the unit circle.
The program I wrote iterates, calculating approximate values the RiemannCircle Taylor series based on an approximation function for sexp(z). And then uses the Taylor series for the RiemannCircle function to calculate another better approximation for the sexp(z) function.
Many, many, many more details to follow! Be patient. This may take a few days....
- Enjoy, Sheldon
This thead contains some of the mathematical equations I used for the fast Kneser algorithm. In this post, B is the base for the sexp function, and L is the fixed point for base B.
This is the complex valued superfunction, developed from the fixed point L for base B, where B>\( \eta \)
\( c=L\times\ln(B) \) for base e, c=L
\( \operatorname{superf}_{B}(z) = \lim_{n \to \infty}
B^{[n](L + c^{z-n})} =
\lim_{n \to \infty} \exp^{[n]} ((L+c^{z-n})/\ln(B)) \)
This is the complex valued inverse superfunction, developed from the fixed point, which is the inverse of the equation above. The inverse superfunction has the property, that isuperf(B^z)=isuperf(B)+1. This particular equation is normalized, so that it converges to the same value as the limit of n approaches infinity. Both of these two functions are implemented in the pari-GP program I wrote.
\( \operatorname{isuperf}_{B}(z) = \lim_{n \to \infty}
\log_{c} (c^n \times ((\log_{B}^{[n]}(z))-L)) \)
If we started with a pefect sexp(z) function, then this is the 1-cyclic theta function linking the sexp with the superf/isuperf.
\( \theta(z)=\operatorname{isuperf}(\operatorname{sexp}(z))-z \)
\( \operatorname{sexp}(z)=\operatorname{superf}(z+\theta(z)) \)
Theta(z) has a singularity at all integer values of n. Theta(z) is represented by an infinite sequence of fourier terms. The fourier series for theta(z) can be developed from any arbitrary unit length on the real axis of sexp(z), where z>-2. Only terms with positive values of n are included, and all terms a_n for negative values of n are zero.
\( \theta(z)=\sum_{n=0}^{\infty}a_n\times \e^{(2\pi n i z)} \)
Theta(z) is intimately connected to the Riemann unit circle mapping, used by Kneser's construction. The Taylor series for the Riemann unit circle function (I'm not sure of the correct notation here) uses the exactly the same a_n coeffecients as the 1-cyclic theta function! This is something that connects the complex fourier analysis of theta(z) to the theory of complex analytic functions, which is really neat! The RiemannCircle has a singularity at z=1, which corresponds to the singularities at the integer values of theta(z).
\( \operatorname{RiemannCircle}(z) = \theta(\ln(z)/2\pi i) \)
\( \theta(z) = \operatorname{RiemannCircle}(e^{2\pi i z}) =\operatorname{isuperf}(\operatorname{sexp}(z))-z \)
\( \operatorname{RiemannCircle(z)=\sum_{n=0}^{\infty}a_n\times z^n \)
If we had a perfect sexp(z) Taylor series, then we have a function for the values of the Riemann unit circle function, which is generated from the theta function, using the equation above, from the inverse superfunction. Now, we can use Cauchy's integral formula to calculate the Taylor series for the Riemann unit circle function. And this also gives us the coefficients of the 1-cyclic theta(z) function.
Of course, there is still the problem of the singularity on the unit circle, which causes problems due to slow convergence. In later posts, I will try to go into some detail, showing values for the Taylor series results for the Riemann circle function, and how the coefficients slowly decay, with poor convergence on the unit circle.
The program I wrote iterates, calculating approximate values the RiemannCircle Taylor series based on an approximation function for sexp(z). And then uses the Taylor series for the RiemannCircle function to calculate another better approximation for the sexp(z) function.
Many, many, many more details to follow! Be patient. This may take a few days....
- Enjoy, Sheldon
