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 Tetration extension for bases between 1 and eta dantheman163 Junior Fellow Posts: 13 Threads: 3 Joined: Oct 2009 11/05/2009, 03:00 AM (This post was last modified: 11/05/2009, 09:14 PM by dantheman163.) I believe I have found an analytic extension of tetration for bases 1 < b <= e^(1/e). This is based on the assumption (1) The function y=b^^x is a smooth, monotonic concave down function Conjecture: If assumption (1) is true then ${}^x b = \lim_{k\to \infty} (log_{b}^{ok}(x({}^k b- {}^{(k-1)} b)+{}^k b) )$ for $-1 \le x\le 0$ Some properties: This formula converges rapidly for values of b that are closer one. For base eta it converges to b^^x for all x but this is not true for the other bases. Interestingly for b= sqrt(2) and x=1 it seems to be converging to the super square root of 2 I will try to post a proof in the next couple of days I just need some time to type it up. Thanks « Next Oldest | Next Newest »

 Messages In This Thread Tetration extension for bases between 1 and eta - by dantheman163 - 11/05/2009, 03:00 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/05/2009, 01:44 PM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 11/05/2009, 11:53 PM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/07/2009, 09:31 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/07/2009, 05:11 PM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/07/2009, 08:12 PM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 11/07/2009, 11:30 PM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/08/2009, 02:44 PM RE: Tetration extension for bases between 1 and eta - by mike3 - 11/12/2009, 07:11 PM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/15/2009, 01:01 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 12/15/2009, 01:40 AM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/15/2009, 01:48 AM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/17/2009, 02:40 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 12/17/2009, 10:59 AM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/19/2009, 05:06 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 12/19/2009, 10:55 AM

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