Status of proofs tommy1729 Ultimate Fellow Posts: 1,372 Threads: 336 Joined: Feb 2009 07/20/2010, 10:02 PM (08/29/2007, 05:04 AM)andydude Wrote: 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. 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: $ D = \left\{ (b, x) \text{ where } b > 0 \text{ and } \begin{cases} x > -1 & \text{ if } b = 1, \\ x > -2 & \text{ otherwise}. \end{cases} \right\}$ 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 the links dont work andrew. ( message : forbidden ) so i cant understand what you meant. bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 07/21/2010, 03:08 AM (07/20/2010, 10:02 PM)tommy1729 Wrote: the links dont work andrew. ( message : forbidden ) so i cant understand what you meant. As a side note: That's why I always encourage posters to upload pictures instead of just referencing. One never know how long the sites out there will live, even if it is your own personal site. However this forum is intended to survive for a long time. tommy1729 Ultimate Fellow Posts: 1,372 Threads: 336 Joined: Feb 2009 07/21/2010, 10:39 PM for a long time ? will your children take over your role if neccessary ? i hope that wont be needed in the near future. « Next Oldest | Next Newest »