[UFO] Attracting Fixpoints or attracting line?
#13
Hmm, perhaps it is interesting to add some context to that whole idea.

I was playing with two other aspects of iteration of the exponential.


First aspect is the spiraling of the orbit of a complex initial value in the complex plane when repeatedly exponentiated. This suggests, that a polar form of representation of the complex numbers could be more enlightening for the generalization to fractional iterations. And especially, a polar form with a fixpoint as origin.
Indeed this has some charme, even more if one introduces the log-polar-form for a complex number:

\( x = a + b*i = [ \lambda , \phi] \) where \( \lambda = \log(abs(x)) \) and \( \phi \) the arc of the angle.

Then multiplication and division of two values x and y is simply reflected by addition/subtraction of the log-polar components, k'th iterated multiplication (powers) just multiplication of the components by k and exponentiation is \( x_1 = \exp(x_0) \equiv [\lambda _1, \phi _1] = e^{\lambda _0}*[\cos(\phi _0), \sin(\phi _0)] \) and the logarithm results from the inverse of this operation.

Fractional iterates of complex initial numbers look then like a very smooth and even nearly linear interpolation of that lengthes and arcs - but only on first glance: the spiral of the flow is distorted by something which looks a bit like a perihel-effect. It diminuishes (and the flow smoothes) when the fixpoint is approached and extends/chaotizes(?) when the fixpoint is left by iteration.

But leave this here as just another curious observation. More in the focus was the log-polar-representation itself: here the second component (the arc of the angle) exceeds very naturally the interval 0..2*Pi once we introduce iterated addition and multiplication on the log-polar-coefficients and operations back and forth can be meaningfully concatenated without loss of information.

This leads then to the ...
... second aspect, the idea of the introduction of a "winding-number".

Once the idea (and the verbal term) is there, you find related material online; for instance a couple of very good articles of R.M.Corless et al (see below) where they just deal with that winding-number and its possible significance for an extended representation of complex numbers. In the above log-polar-representation we see it nicely that the complex arithmetic via exponentials and logarithms is just somehow modular to modulus 2*Pi (in the arc-parameter) and it is suggestive to introduce a representation for complex numbers and complex algebra which leaves the modular arithmetic behind. It is really nice to see this when playing around a bit. (The observation of "fix-points" vs. "fix-line" as described in this thread was just a detail in the context of exercising in that framework).

However, as also Corless et al discuss in their article: by buying simplicity for multiplication and exponentiation using the log-polar-form we pay with complication for addition. Actually, Corless et al, as I understood them, gave up because of intractability of addition when the winding-number is introduced.

That was also my idea and also I give up here.

We might extend the arc-parameter phi, possibly rescale it to the 2Pi-unit and then allow not only the interval (modulo 1) but the full real line: then a/the "winding-number" is represented by the integer part. But this does not help much: as long as we have no meaningful interpretation of the addition of two log-polarforms, where the winding-number is kept significant such that we can, for instance, invert the operations.
I felt, it would really be a nice toy-project to experiment with this a bit, however... my time (and energy) is somehow more limited than in the previous years.

Gottfried

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


Corless, R. M., and Jeffrey, D. J.
The unwinding number.
Sigsam Bulletin 30, 2 (June 1996), 28-35.

[online-version:
http://www.apmaths.uwo.ca/~djeffrey/Offp...ditors.pdf
Editor's Corner: The Unwinding Number
]


Corless, R. M., and Jeffrey, D. J.
The Wright omega function
Ontario Research Centre for Computer Algebra and the Department of Applied Mathematics
University of Western Ontario, London, Canada
Reference: J. Calmet et al (Eds.): AISC-Calculemus 2002, LNAI 2385, pp. 76-89, 2002, Springer-Verlag


Reasoning about the elementary functions of complex analysis
Corless, R.M., Jeffrey, D.J. and Davenport, J.H.
[can be found online]
Abstract:
There are many problems with the simplification of elementary functions, particularly over the complex plane. Systems tend to make “howlers” or not to simplify enough. In this paper we outline the “unwinding number” approach to such problems, and show how it can be used to systematise such simplification, even though we have not yet reduced it to a complete algorithm. The unsolved problems are probably more amenable to the techniques of artificial intelligence and theorem proving than the original problem of complex-variable analysis.
Keywords: Elementary functions; Branch cuts; Complex identities.
Gottfried Helms, Kassel


Messages In This Thread
RE: [UFO] Attracting Fixpoints or attracting line? - by Gottfried - 04/09/2010, 08:30 PM

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