Hi Henryk -
I see. And I also see that I've to undergo a private derivative-workshop
soon. When I re-read in our forum last days I (again) found so many things which I'd missed in time when they were posted, - because I was too much consumed by the questions I had to solve from my own approach, and had no other space...
Yes, in the first msg I had only a "bunch-of-numbers"-series for which I claimed, it would provide the tetrate of sqrt(2) given the height h in the u^h-parameter. It's just a normed form of the inverse schröder-function sigma°-1 which occurs in the diagonalization. Not much finding, anyway, just to mention for the record.
In the second post I had the idea to decompose the coefficients of the series into some simple components: the reciprocals of natural numbers, the second-powers of u. and a remainder (for each coefficient).
Now the series looks no more like an anonymous object, but with a clear property, that the so-found remainders converge to the reciprocal of the log(b) :
\( \lim_{k->\infty} r_k = \frac{1}{\log(b)} \)
where r is the remainder as described above
The terms as given in the first msg can thus be decomposed in this form (for k>0)
\( c_k*(u^{h})^k = \frac{r_k}{k}*(u^{2+h})^k \)
or, using coefficients a, where the log(b) -part is also extracted
\( b\^\^ ^h = t - \frac{\sum_{k=1}^{\infty} \frac{a_k}{k}*(u^{2+h})^k }{\log(b)} \)
Here, according to the visual impression, 0 < a_k <1 approching a_k->1 for k->inf, and, in a second view, even strictly increasing
Now two considerations:
a) what does it mean, if the visually apparent convergence actually exists? Well, it means that in the limit h-> -2 we would get the zeta(1)-singularity, which can also be written differently as log(eps), for eps->0 and the interesting limit-identity at h=-2 including the two infinite expressions and the fixpoint.
b) This is then interesting - and I asked: does such an identity(in the limit, though) make sense/exists at all? It does/exists, as our discussion show.
From this, in turn, the series in its decomposed form, the remainders as coefficients, seems to "implement" just this interesting identity for the limit case. and gets then from this a special justification.
The series shows the coefficients for the base b=sqrt(2). But as it is just taken from the (normed) Schröder-inverse, the same can be said for other bases for 1<b<eta for which we'll get a modified series, but with the same property of limit at h=-2 and the convergence of the remainders.
Well, I think I'll append a description in the "matrix-operator-method"-thread, maybe tonight, maybe tomorrow (health is badly degrading this weeks/monthes, cannot work/concentrate as long as was usual in the previous years, sorry)
Gottfried
bo198214 Wrote:We dont get it, except from a specific tetration (while the formula I gave is universal to all tetrations).
However if we have the derivation at 0 then we have it also at -1 and all natural numbers (by the chain rule).
I see. And I also see that I've to undergo a private derivative-workshop
soon. When I re-read in our forum last days I (again) found so many things which I'd missed in time when they were posted, - because I was too much consumed by the questions I had to solve from my own approach, and had no other space...Quote:I dont even know what series you are talking about, I just saw a bunch of floating numbers that are said to have something to do with the matrix/diagonalization approach.
Yes, in the first msg I had only a "bunch-of-numbers"-series for which I claimed, it would provide the tetrate of sqrt(2) given the height h in the u^h-parameter. It's just a normed form of the inverse schröder-function sigma°-1 which occurs in the diagonalization. Not much finding, anyway, just to mention for the record.
In the second post I had the idea to decompose the coefficients of the series into some simple components: the reciprocals of natural numbers, the second-powers of u. and a remainder (for each coefficient).
Now the series looks no more like an anonymous object, but with a clear property, that the so-found remainders converge to the reciprocal of the log(b) :
\( \lim_{k->\infty} r_k = \frac{1}{\log(b)} \)
where r is the remainder as described above
The terms as given in the first msg can thus be decomposed in this form (for k>0)
\( c_k*(u^{h})^k = \frac{r_k}{k}*(u^{2+h})^k \)
or, using coefficients a, where the log(b) -part is also extracted
\( b\^\^ ^h = t - \frac{\sum_{k=1}^{\infty} \frac{a_k}{k}*(u^{2+h})^k }{\log(b)} \)
Here, according to the visual impression, 0 < a_k <1 approching a_k->1 for k->inf, and, in a second view, even strictly increasing
Now two considerations:
a) what does it mean, if the visually apparent convergence actually exists? Well, it means that in the limit h-> -2 we would get the zeta(1)-singularity, which can also be written differently as log(eps), for eps->0 and the interesting limit-identity at h=-2 including the two infinite expressions and the fixpoint.
b) This is then interesting - and I asked: does such an identity(in the limit, though) make sense/exists at all? It does/exists, as our discussion show.
From this, in turn, the series in its decomposed form, the remainders as coefficients, seems to "implement" just this interesting identity for the limit case. and gets then from this a special justification.
The series shows the coefficients for the base b=sqrt(2). But as it is just taken from the (normed) Schröder-inverse, the same can be said for other bases for 1<b<eta for which we'll get a modified series, but with the same property of limit at h=-2 and the convergence of the remainders.
Quote:I would really appreciate if you could be more clear in your terminology:
The matrix power approach makes use of the established method to obtain non-integer powers (and other analytic functions) of finite matrices via diagonalization.
This is applied to the truncations of the Carleman/Bell matrix.
Well, I think I'll append a description in the "matrix-operator-method"-thread, maybe tonight, maybe tomorrow (health is badly degrading this weeks/monthes, cannot work/concentrate as long as was usual in the previous years, sorry)
Gottfried
Gottfried Helms, Kassel