Am 01.02.2008 10:27 schrieb se16@[EMAIL PROTECTED]
> On 1 Feb, 09:05, James Dow Allen <jdallen2...@[EMAIL PROTECTED]
> wrote:
>> e+e+e+e+...+e (N times) is e*N
>> and can be defined for non-integer N
>> e*e*e*e*...*e (N times) is e^N
>> and can be defined for non-integer N
>> What is e^e^e^...^e (N times) ?
>> Can it be defined for non-integer N?
>
> It depends on whether you think a^b^c is
> (a) (a^b)^c or (b) a^(b^c)
>
> In case (a) e^e^e^...^e (N times) is e^(e^(N-1)) which is easily
> extended to non-integer N
>
> In case (b) - more usual since it is more interesting - the you are
> looking at "tetration". It can be extended to non-integer N in
> several different ways. See
http://en.wikipedia.org/wiki/Tetration#Extension_to_real_heights
There are some approaches to that problem.
Some of them use powerseries-expansion and modified
exponential-series.
The exponential series for non-integer heights must
be found by "fixpoint"-analysis z = e^z => e = z^(1/z)
z = h(e) where h() is a variant of the lambert-w-function.
There are multiple (complex) such fixpoints for e.
In a first approach it seems, that using any of that
fixpoints should produce the same result; however it
was re****ted (only a statement, no article) that the
results are different for different fixpoints (I only have
a second-hand-remark in the "tetration-forum" (*1), see
thread "bummer").
If this is indeed so, then the whole idea of fractional
iterations is much more complicated than expected, and
possibly not consistent without explicite selection of
a specific fixpoint.
(We guys in the tetration-forum are occasionally
working on this, but don't have a more precise description
of the problem yet, and no solution as well.)
Gottfried
(*1) http://math.eretrandre.org/tetrationforum/index.php
--
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Gottfried Helms, Kassel
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