Re: tetration and logaritms

Am 10.07.2008 10:18 schrieb lwalke3@xxxxxxxxx:
On Jul 9, 11:59 am, amy666 <tommy1...@xxxxxxxxxxx> wrote:

And this of course is the well-known Taylor series for log(1+x)
that tommy1729 already mentioned in his post. Finally, h=-1/2:

U_e(x,-1/2) = x-x^2/4+5x^3/48-5x^4/96+109x^5/3840


:)

a) first another word on the matrix-method: if you remove
the "x" while keeping track for same powers of x by arranging
the coefficients in indexed columns according to the power
of x ... then you have a matrix (of coefficients).
This is then simply "my" matrix-method..(I collect like powers
of x in rows, but that doesn't matter)


b) Interpolation: in a previous post I discussed a likely
difference of methods, when different interpolation-approaches
depending on the h-parameter are assumed.
You gave a polynomial interpolation approach, which I think is
somehow natural.
But the coefficients at -for instance- x^1 with increasing h
(1,1,1,1,1,...)
can also be interpolated by -for instance- a sine-function of h,
as well as the coefficients at higher powers of x.

I don't mean to play a game of obfuscation here: the reason
for my being "not-completely-satisfied" is, that -using U-tetration,
as application of T-tetration with fixpoint-shift- the
common tetration (T-tetration in my wording) seems to be dependent
on the selection of the fixpoint, if fractional iterations are computed.

So - while the matrix-based (diagonalization) method using
U-tetration for each fixed base only may be consistent, when
the polynomial interpolation-approach is applied, then still
the cross-base-relations might be "imperfect"...

Hmm - I must be vague this way, because I still don't have
hard data at hand to see these differences (they are said
to be small, may be smaller than my approximation-accuracy)
and so seem currently to be too small to be able to experiment
with this problem effectively.

At least there is one consolidation: in my previous post I
mentioned the different interpolation-method using the
binomial-expansion and values of the powertower-function
themselves (as can be seen for instance in [1],[2] or [3])
Here I found different results in my first comparision
(using insufficient approximation) - however, a new computation
indicates now, that the results of this method and of the
diagonalization may come out to be the same (as always
expected) [4]

Gottfried Helms

------------

[1] Comtet, Louis; Advanced Combinatorics,

[2] Woon, S.C.; Analytic Continuation of Operators —
Operators acting complex s-times
Chap 9 (online available in arXiv-org)

[3] Robbins, Andrew; (forum-message binomial-method=Woon-method)
http://math.eretrandre.org/tetrationforum/showthread.php?tid=186&pid=2319#pid2319

[4] Helms, Gottfried; (binomial-method approximative equal to diagonalization)
http://math.eretrandre.org/tetrationforum/showthread.php?tid=186&pid=2321#pid2321
.

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