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Status of proofs
#9
Well, I have given this a lot of thought, and I believe that it is easier to prove real-analytic tetration than complex-analytic tetration. The reason for this is that the real-valued tetration has a smaller domain than complex-valued tetration. The first realization is that tetration over real numbers can produce complex numbers. After this realization we can eliminate a great deal of the domain over which real-analycity must fail (because it is not continuous, and if its not continuous it can't be analytic). To show what I mean by this I have included a color-coded plot of the log(abs(b^^x)) where gray is a finite real number, blue is a complex number output, and red is indeterminate.

[Image: tetration_domain.png]

I have also included a pretty-version of the domain where the circles indicate indeterminate outputs. The dark-gray quarter-plane is that largest domain over which real-analytic tetration can be defined. The medium-gray are regions which have real outputs, but the dotted line indicates a discontinuity, so this would make real-analycity fail if this were in the domain of real-tetration. The light-gray region is probably not real-valued, but it is real-valued with a first-approximation tetraiton (linear critical).

A mathematical definition of the dark-gray domain is:


So to summarize, I believe that what needs to be proven is that Tetration whose domain is the dark-gray region given above is real-analytic in both b and x. Once it is proven that real-analytic tetration exists over this domain, then we can worry about its uniqueness.

Andrew Robbins
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Messages In This Thread
Status of proofs - by Daniel - 08/24/2007, 06:56 PM
RE: Status of proofs - by bo198214 - 08/24/2007, 07:16 PM
RE: Status of proofs - by bo198214 - 08/24/2007, 07:23 PM
RE: Status of proofs - by Daniel - 08/24/2007, 11:09 PM
RE: Status of proofs - by bo198214 - 08/25/2007, 12:07 AM
RE: Status of proofs - by bo198214 - 08/27/2007, 08:54 AM
RE: Status of proofs - by Gottfried - 08/24/2007, 10:06 PM
RE: Status of proofs - by Daniel - 08/24/2007, 11:02 PM
RE: Status of proofs - by Gottfried - 08/29/2007, 06:07 AM
RE: Status of proofs - by andydude - 08/29/2007, 05:04 AM
RE: Status of proofs - by tommy1729 - 07/20/2010, 10:02 PM
RE: Status of proofs - by bo198214 - 07/21/2010, 03:08 AM
RE: Status of proofs - by tommy1729 - 07/21/2010, 10:39 PM



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