10/11/2021, 03:51 AM
(This post was last modified: 10/11/2021, 02:30 PM by sheldonison.)
Hi James,
Just an update. I've been playing with beta speedup optimizations of about 3 orders of magnitude speedup, and what speeds up beta speeds up everything else. This lets me graph the resultant tetration function pretty easily, and show where it misbehaves etc, and easily generate accurate values for large numbers of derivatives.
I also wanted to go back to my original approximation for where the zeros of \( f(z)=\beta(z)-\ln(1+\exp(-z)) \) are, which is approximately where
\( \beta(z-1)+z=2n\pi i \)
and more rigorously justify the approximation that f(z) has a zero nearby and also that the limiting function \( \lim_{m\to \infty}f^{\circ m}(z)=\ln^{\circ m}(f(z+m)) \) also a a zero nearby. I made some progress making these two approximations a little more rigorous. edit: The graph below is of \( f^{\circ3}(z) \); another interesting graph would be \( f^{\circ3}(z)-f^{\circ2}(z) \).
I also made progress in handling the most "correct" logarithmic branch, which makes possible graphs of the resulting Tet(z) function near Tet(0) which has a "pseudo" radius of convergence of 0.489, based on zero #6 of f(z) as the closest zero of f(z) causing misbehavior in the Tet(z). The zero is at f(5.11154320063377 + 0.324420060794418*I), and tet is centered at ~= beta(1.7448 ). There are also an infinite number of other singularities closer to the origin for larger values of z. This is at a radius=0.5, where the logarithmic singularities cause the sawtooth behavior in this graph.
Just an update. I've been playing with beta speedup optimizations of about 3 orders of magnitude speedup, and what speeds up beta speeds up everything else. This lets me graph the resultant tetration function pretty easily, and show where it misbehaves etc, and easily generate accurate values for large numbers of derivatives.
I also wanted to go back to my original approximation for where the zeros of \( f(z)=\beta(z)-\ln(1+\exp(-z)) \) are, which is approximately where
\( \beta(z-1)+z=2n\pi i \)
and more rigorously justify the approximation that f(z) has a zero nearby and also that the limiting function \( \lim_{m\to \infty}f^{\circ m}(z)=\ln^{\circ m}(f(z+m)) \) also a a zero nearby. I made some progress making these two approximations a little more rigorous. edit: The graph below is of \( f^{\circ3}(z) \); another interesting graph would be \( f^{\circ3}(z)-f^{\circ2}(z) \).
I also made progress in handling the most "correct" logarithmic branch, which makes possible graphs of the resulting Tet(z) function near Tet(0) which has a "pseudo" radius of convergence of 0.489, based on zero #6 of f(z) as the closest zero of f(z) causing misbehavior in the Tet(z). The zero is at f(5.11154320063377 + 0.324420060794418*I), and tet is centered at ~= beta(1.7448 ). There are also an infinite number of other singularities closer to the origin for larger values of z. This is at a radius=0.5, where the logarithmic singularities cause the sawtooth behavior in this graph.
- Sheldon