Just came across an older subject and thought it would fit into this "norming"-thread.
As older fellows here may remember, nearly my first contact with tetration was the question of alternating iteration-series for which I worked out some interesting heuristics. (see [1] and [2])
[update] I should explain, that for convenient ascii-notation of the tetration I "misuse" here the common notation. With z^^h I mean in the context of a given fixed base b, the value of z^^h := \( \exp_b^{oh}(z) \) [/update]
Using base b=sqrt(2) we have the real-valued interval 2..4 for which we may find iteration heights from -inf to +inf if we start at some value z, say z=3, in this interval. Because in both direction the values of z^^h are finite we can compute a value for the alternating series of that values. So using Pari/Gp we can compute
f(z) = sumalt(h=0,(-1)^h*iter(z,h)) + sumalt(h=0,(-1)^h*iter(z,-h) ) - z
to evaluate the alternating iteration-series with center at the chosen z.
It is clear that this series is periodic for z in the interval z..z^^2 . But what's interesting is, that in general the f(z) is "small" and even we find f(z)=0
Because this is a remarkable result (and matches, for instance, the analogue problem when applied to a doubly-infinite geometric series by analytic continuation) this value z (where f(z)=0) introduces itself gently as candidate for a norm-value, at which the height is defined to be zero or at least an integer.
Here is a picture of the sinusoidal curve f(z) when z is moved from z to z^^2 beginning at some arbitrary value z0:
We see that astonishing approximation to a sine-curve, where the amplitude should be normed. Actually the deviance from the sine-curve is of the order of 1e-3 : I mean, if the height-parameter of this curve is compared with the abscissa of the sine-curve after the two curves are matched (for instance by binary search of the same y-values).
I'm not experienced with Fourier-analysis, but I think, it would be profitable to try to describe the f(z)-function by a fourier-decomposition. Analoguously this could be done for the other bases 1<b<eta.
Gottfried
[update]: obviously this provides also a "fixpoint-independent" definition for the real fractional tetration: just match the values of the sin-curve with that of f(z) and define the height h for the representation of the z according to the found abszissa of the sine (though this provides only approximation). [/update]
(both articles are *very* freshman-like and need being improved...)
[1] Short article of magazine-type
[2] longer version
As older fellows here may remember, nearly my first contact with tetration was the question of alternating iteration-series for which I worked out some interesting heuristics. (see [1] and [2])
[update] I should explain, that for convenient ascii-notation of the tetration I "misuse" here the common notation. With z^^h I mean in the context of a given fixed base b, the value of z^^h := \( \exp_b^{oh}(z) \) [/update]
Using base b=sqrt(2) we have the real-valued interval 2..4 for which we may find iteration heights from -inf to +inf if we start at some value z, say z=3, in this interval. Because in both direction the values of z^^h are finite we can compute a value for the alternating series of that values. So using Pari/Gp we can compute
f(z) = sumalt(h=0,(-1)^h*iter(z,h)) + sumalt(h=0,(-1)^h*iter(z,-h) ) - z
to evaluate the alternating iteration-series with center at the chosen z.
It is clear that this series is periodic for z in the interval z..z^^2 . But what's interesting is, that in general the f(z) is "small" and even we find f(z)=0
Because this is a remarkable result (and matches, for instance, the analogue problem when applied to a doubly-infinite geometric series by analytic continuation) this value z (where f(z)=0) introduces itself gently as candidate for a norm-value, at which the height is defined to be zero or at least an integer.
Here is a picture of the sinusoidal curve f(z) when z is moved from z to z^^2 beginning at some arbitrary value z0:
We see that astonishing approximation to a sine-curve, where the amplitude should be normed. Actually the deviance from the sine-curve is of the order of 1e-3 : I mean, if the height-parameter of this curve is compared with the abscissa of the sine-curve after the two curves are matched (for instance by binary search of the same y-values).
I'm not experienced with Fourier-analysis, but I think, it would be profitable to try to describe the f(z)-function by a fourier-decomposition. Analoguously this could be done for the other bases 1<b<eta.
Gottfried
[update]: obviously this provides also a "fixpoint-independent" definition for the real fractional tetration: just match the values of the sin-curve with that of f(z) and define the height h for the representation of the z according to the found abszissa of the sine (though this provides only approximation). [/update]
(both articles are *very* freshman-like and need being improved...)
[1] Short article of magazine-type
[2] longer version
Gottfried Helms, Kassel
