Re: Could a tetration function exist?

On Nov 24, 8:31 pm, lwal...@xxxxxxxxx wrote:
On Nov 24, 1:17 pm, mike3 <mike4...@xxxxxxxxx> wrote:





Another question I had was this. The domain in which
the infinite power tower ^oo x converges on the reals
is between e^(-e) and e^(1/e), inclusive. But what
about on the complex numbers?
So then it would seem that ^4 b would be very
close to zero, as an analysis with Euler's formula
would suggest: The real part gives a value for the
log of the magnitude of the complex number. Since
the log is a giant negative number, the magnitude
must be incredibly close to zero (too close to write
down even if you had a billion billion billion
trillion trillion trillion universes with all their
matter turned to pens, ink, and paper.). This
suggests the sequence actually converges, but only
after swinging up to a silly huge number. Does this
mean tetration converges on the whole complex plane
except for the part of the real line outside
[e^(-e), e^(1/e)]?

Another tetration expert, Daniel Geisler, has already
considered this question:

http://www.tetration.org/Fractals/Atlas/index.html

"The mastery of tetration requires the application of the
theoretical to the explaination of the empirical. The goal
of this atlas is to catalog the major structures of the
tetration fractal. The most interesting features are a
esult of ^n x taking very small or large values where x is
a real number, typically a negative real number and n is
a whole number. The nature of tetration is such that
usually if ^n z has an extremely large magnitude then
often a small value of epsilon can be found such that
^(n-1)(z+epsilon) ~ -^(n-1)z forcing ^n(z+epsilon) to a
ery small value and giving ^(n+1)(z+epsilon) ~ 1 and
^(n+2)(z+epsilon) ~ z+epsilon."

You have discovered that the tetration of 6 + 4i is
eventually 5-periodic. Geisler has graphed a fractal
showing the regions where tetration is n-periodic, for
various values of n ("tetration by period"). Here where
n=1, the tetration actually converges. He also shows the
region where it fails to converge ("tetration by escape").

So I guess this isn't all that new, then. <sigh>

But I'm curious. What are those strange "finger" things in
the fractal?
.

Relevant Pages

  • !! tetration problem (abel-taylor problem) !!
    ... While trying to find a solution to tetration one often tries Abel-like equations. ... (not usually called abel but see remark below how this can be turned into abel and is thus basicly somewhat equivalent) ... But exp))) = q has complex solutions q, thus those do not satisfy the equation. ... thus fis not analytic in any neighbourhood of the reals. ...
    (sci.math)
  • Re: Could a tetration function exist?
    ... the infinite power tower ^oo x converges on the reals ... log of the magnitude of the complex number. ... trillion trillion trillion universes with all their ... Another tetration expert, Daniel Geisler, has already ...
    (sci.math)
  • Re: Could a tetration function exist?
    ... of tetration. ... Uh, yes, I said in the post I was limiting this to reals x> 1 and ... The iteration of entire transcendental ... f)=F, Math. ...
    (sci.math)
  • Re: Bringing back an old tetration curiosity?
    ... Both estimates bound the order-2 tetraroot from below. ... In general I don't expect ANY calculable tetration function to pass "through" ... Any tetration extension to the reals is like a mathematical ...
    (sci.math)
  • Re: Could a tetration function exist?
    ... Andrew Robbins and Gottfried Helms, ... base of the power tower is the "base" of the tetration! ... trillion trillion trillion universes with all their ...
    (sci.math)
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