01/02/2010, 05:31 AM
(This post was last modified: 01/02/2010, 02:05 PM by sheldonison.)

Mike, thanks for your reply. At this point, I'm actually interested in the behavior of analytic tetration, and I'm using the base conversion equations and cheta to probe the behavior of the analytic super exponentials, (more below).

Inherent ringing in analytic tetration

The pattern I like the most is simply that the analytic super-exponentials for different bases can criss cross each other an infinite number of times, as they grow super-exponentially towards infinity. If you line up cheta with another super-exponential function, at cheta(4.884+n), then they will cross an infinite number of times. For integer values of n>=0, and any base b>, if k and x are chosen such that x=4.884+n, and such that

than it turns out that for every base b>, there is a pattern in the first derivative, that causes the two super-exponential functions to wind around each other as they both grow super-exponentially towards infinity.

The sexp_b(x) function is growing slower than cheta for all bases>, in the region of cheta(4.884+n). Approximately 1/2 of an iterate greater, there will be another crossing where cheta(x+0.5) will equal sexp_b(x+k+0.5). In that region,

.

Here is a graph, that hopefully shows the pattern. For several bases, I am graphing the analytic slog_b of cheta(x), where the two super-exponentials have been lined up as described above. The amplitude gets smaller as the base approaches , but the windings are always there. This is also a little disturbing, in that this is the analytic extension of tetration to real numbers, and has nothing to do with base conversions from cheta to other bases. Different bases have this bizarre winding pattern when the different bases are lined up and compared against each other. Also, the phase of the winding pattern is somewhat constant. It would be interesting to try to prove this pattern, but I haven't made any progress on that yet.

I chose to line up the different super exponentials at cheta(5.884), since that is close to the 50% duty cycle for bases with b approaching , with crossings near cheta(5.884+n) and cheat(5.384+n). At base e, the actual "50%" line is around cheta(5.878+n), cheta(5.378+n). There is a small shift as the base increases, but I don't have access to an accurate enough sexp function for bases>1.6, except for base e. For x<=4.11, the other super-exponentials are all larger than cheta(x), , with cheta(x) first growing larger than the other super-exponentials between cheta(4.12) and cheta(4.14). Cheta(4.12) is approximately , or near a googol. For cheta(x), with x>5, the pattern converges rapidly to a 1-cyclic function.

(12/31/2009, 11:45 PM)mike3 Wrote: The graph I gave for was done via the Cauchy integral. It should be possible also to use the Cauchy integral at other bases greater than . I'll see if I could try one for to get a graph and Taylor series approximation.Did you generate the Cauchy integral, or use Kouznetsov's published Taylor series, (which were generated from the cauchy integral)? That would be neat to publish values for the Taylor series for other bases. I wasn't totally clear on what his method was, even after reading the published pdf file. It sounds like he needed a "seed" approximation which was then improved by using the Cauchy integral.

Quote:I do wonder though, even if cannot be used to approximate , whether it is still possible that maybe and can be used to approximate each other, for real and greater than . However, the fine detail in that fractal thingy in the graph makes it seem questionable.You switch one problem for another. Instead of generating the Taylor series, you need to generate the Fourier series terms. But the Fourier series is real valued, and doesn't give you the complex plane. There are some patterns for the fourier series coeffients of for different bases. For example, the 2nd overtone, k2*sin(4*pi*x+b2) coefficient seems to always be approximately 100 times smaller than the primary primary coefficient, k1*sin(2*pi*x+b1). Also, the phase terms, b1 and b2, appear to be relatively constant.

Inherent ringing in analytic tetration

The pattern I like the most is simply that the analytic super-exponentials for different bases can criss cross each other an infinite number of times, as they grow super-exponentially towards infinity. If you line up cheta with another super-exponential function, at cheta(4.884+n), then they will cross an infinite number of times. For integer values of n>=0, and any base b>, if k and x are chosen such that x=4.884+n, and such that

than it turns out that for every base b>, there is a pattern in the first derivative, that causes the two super-exponential functions to wind around each other as they both grow super-exponentially towards infinity.

The sexp_b(x) function is growing slower than cheta for all bases>, in the region of cheta(4.884+n). Approximately 1/2 of an iterate greater, there will be another crossing where cheta(x+0.5) will equal sexp_b(x+k+0.5). In that region,

.

Here is a graph, that hopefully shows the pattern. For several bases, I am graphing the analytic slog_b of cheta(x), where the two super-exponentials have been lined up as described above. The amplitude gets smaller as the base approaches , but the windings are always there. This is also a little disturbing, in that this is the analytic extension of tetration to real numbers, and has nothing to do with base conversions from cheta to other bases. Different bases have this bizarre winding pattern when the different bases are lined up and compared against each other. Also, the phase of the winding pattern is somewhat constant. It would be interesting to try to prove this pattern, but I haven't made any progress on that yet.

I chose to line up the different super exponentials at cheta(5.884), since that is close to the 50% duty cycle for bases with b approaching , with crossings near cheta(5.884+n) and cheat(5.384+n). At base e, the actual "50%" line is around cheta(5.878+n), cheta(5.378+n). There is a small shift as the base increases, but I don't have access to an accurate enough sexp function for bases>1.6, except for base e. For x<=4.11, the other super-exponentials are all larger than cheta(x), , with cheta(x) first growing larger than the other super-exponentials between cheta(4.12) and cheta(4.14). Cheta(4.12) is approximately , or near a googol. For cheta(x), with x>5, the pattern converges rapidly to a 1-cyclic function.