03/02/2012, 09:20 AM
(This post was last modified: 03/02/2012, 09:36 AM by sheldonison.)

The reason I developed the merged tetration code was to help develop some numerical evidence about the behavior to tetration in the neighborhood of eta, and whether or not eta is a branch point of complex tetration. In that sense, this thread could be viewed as a continuation of eta as a branchpoint of tetrational. From what has been posted thus far, if you consider complex tetration from etaB+r*exp(-Pi*I) ... etaB+r*exp(Pi*I), then as the radius goes to zero, the numerical results would suggest the singularity at eta is very mild, nearly vanishing as you approach eta from different directions.

Yesterday, I posted a sequence of complex plane graphs, of sexp_b(z) as the base rotates around eta. The last graph in the sequence was for base=1.621-0.177i, which is 7/8th of the way around the eta, with a 0.25 radius. This base didn't converge with the default settings. That's because my program assumes the radius of convergence for sexp(z) around z=0 is 2, but for this base, it is more like 1.2 or 1.3. To be doubly sure these results are valid, I increased the precision to 67 decimal digits, and then I had to tripled the number of terms in the sexp(x) taylor series, over what is required by default to get results accurate to 32 decimal digits. I got convergence, accurate to 32 decimal digits, and I'm plotting the results below. There's two plots below. One for base=eta+0.25*exp(1.75 pi i), and one for base=eta+0.25*exp(-0.25 pi i), corresponding to rotating 7/8th of the way around eta, counter clockwise, or rotating -1/8th of the way around eta. The two bases are the same, but the two solutions are very different.

The solution for merged tetration rotating 7/8th of the way around eta is pretty bizarre, and is near the edge of what my algorithm will converge for. A little farther, at 94.4% of the way around the circle is the Shell Thron boundary for a radius of 0.25. At we approach 94.4% of the way around the circle, to get to the Shell thron boundary, the radius of convergence for the merged solution gets arbitrarily small, and at >=90% of the way around the circle, the radius of convergence is probably<1, which breaks the tetcomplex.gp algorithm as currently written. I think the graphs will help show what is happening.

Function 1) complex graph1, 7/8th of the way around the circle

Function 2) complex graph2, -1/8th of the way around the circle

First off, contrast the difference in these two functions, with the near identical similarity with the functions +/-50% of the way around the circle. Function1-Function2, around the unit circle at z=0, differ from each other by as much as 0.03, versus a much smaller difference of <10^-11 for the two functions at eta+0.25*exp(-pi), and eta+0.25+exp(pi), going halfway around the circle clockwise and counterclockwise to eta-0.25. Second, the differences for bases<eta get arbitrarily small as the period gets arbitrarily large for bases approaching eta. That may not be the case when the base approaches the Shell Thron boundary, after rotating most of the way around the circle around eta.

Finally, this solution for 7/8th of the way is pretty bizarre. It is almost like the function is upside down. In the upper half of the complex plane, when imag(z)>0, the function converges to approximately L=1.35-1.07i, but only if real(z) is much larger than imag(z). In the lower half of the complex plane, with imag(z)<0, the function converges to L2 which is approximately 1.614+2.334i, but only when real(z) is very large negative compared to imaginary z.

- Sheldon

This is the bizarre "upside down" solution you get, rotating 7/8th of the way around eta, counter clockwise, with a radius of 0.25, based on Taylor series results generated accurate to 32 decimal digits.

The much more normal solution you get if you rotate -1/8th of the way around eta, with a radius of 0.25

Yesterday, I posted a sequence of complex plane graphs, of sexp_b(z) as the base rotates around eta. The last graph in the sequence was for base=1.621-0.177i, which is 7/8th of the way around the eta, with a 0.25 radius. This base didn't converge with the default settings. That's because my program assumes the radius of convergence for sexp(z) around z=0 is 2, but for this base, it is more like 1.2 or 1.3. To be doubly sure these results are valid, I increased the precision to 67 decimal digits, and then I had to tripled the number of terms in the sexp(x) taylor series, over what is required by default to get results accurate to 32 decimal digits. I got convergence, accurate to 32 decimal digits, and I'm plotting the results below. There's two plots below. One for base=eta+0.25*exp(1.75 pi i), and one for base=eta+0.25*exp(-0.25 pi i), corresponding to rotating 7/8th of the way around eta, counter clockwise, or rotating -1/8th of the way around eta. The two bases are the same, but the two solutions are very different.

The solution for merged tetration rotating 7/8th of the way around eta is pretty bizarre, and is near the edge of what my algorithm will converge for. A little farther, at 94.4% of the way around the circle is the Shell Thron boundary for a radius of 0.25. At we approach 94.4% of the way around the circle, to get to the Shell thron boundary, the radius of convergence for the merged solution gets arbitrarily small, and at >=90% of the way around the circle, the radius of convergence is probably<1, which breaks the tetcomplex.gp algorithm as currently written. I think the graphs will help show what is happening.

Function 1) complex graph1, 7/8th of the way around the circle

Function 2) complex graph2, -1/8th of the way around the circle

First off, contrast the difference in these two functions, with the near identical similarity with the functions +/-50% of the way around the circle. Function1-Function2, around the unit circle at z=0, differ from each other by as much as 0.03, versus a much smaller difference of <10^-11 for the two functions at eta+0.25*exp(-pi), and eta+0.25+exp(pi), going halfway around the circle clockwise and counterclockwise to eta-0.25. Second, the differences for bases<eta get arbitrarily small as the period gets arbitrarily large for bases approaching eta. That may not be the case when the base approaches the Shell Thron boundary, after rotating most of the way around the circle around eta.

Finally, this solution for 7/8th of the way is pretty bizarre. It is almost like the function is upside down. In the upper half of the complex plane, when imag(z)>0, the function converges to approximately L=1.35-1.07i, but only if real(z) is much larger than imag(z). In the lower half of the complex plane, with imag(z)<0, the function converges to L2 which is approximately 1.614+2.334i, but only when real(z) is very large negative compared to imaginary z.

- Sheldon

This is the bizarre "upside down" solution you get, rotating 7/8th of the way around eta, counter clockwise, with a radius of 0.25, based on Taylor series results generated accurate to 32 decimal digits.

The much more normal solution you get if you rotate -1/8th of the way around eta, with a radius of 0.25