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 Tetration below 1 Gottfried Ultimate Fellow Posts: 758 Threads: 117 Joined: Aug 2007 09/03/2007, 08:41 PM bo198214 Wrote:Gottfried Wrote:Quote:However perhaps it could be that ${}^{\frac{1}{2}}b=f^{\circ \frac{1}{2}}(1)$ is real or generally that ${}^tb=f^{\circ t}(1)$ is real, which I dont believe. Can someone just compute it? Hmm, I don't know, whether I understand you correctly. If h=1 then all eigenvalues except the first are zero ( = [1,0,0,....]) and the result is always the same, independent of any power of log(h) since the "height" y of the tower occurs only as exponent of the eigenvalues.... Did I misread something obvious? As said $h$ is not 1 but $e^{-1}. $f(x)=b^x$, $A$ is the power derivation matrix of $f$ (I think this is the transpose of your matrix $B_b$), and $A^t$ is $\exp(t\cdot \log A)$ (though we can also apply the powerseries $(1+x)^t$ directly to $A$). Hence the power series $f^{\circ t}(x)$ has as coefficients the first row of $A^t$ (think transposed in your notation). And now tetration is defined as ${}^tb=f^{\circ t}(1)$. We set $x=1$ not $h$. ok, I misread that $f^{\circ t}(x)$ as x^^t instead of {h,x}^^t, as in Andrew's notational references. I got it now. If there are only two parameters given as in $f^{\circ t}(x)$ I automatically assume, that it is {x,1}^^t instead of {,x}^^t. I think I'll have to get used to it now... Thanks - Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Tetration below 1 - by bo198214 - 08/29/2007, 06:14 PM RE: Tetration below 1 - by Daniel - 08/30/2007, 09:32 PM RE: Tetration below 1 - by jaydfox - 08/30/2007, 11:08 PM RE: Tetration below 1 - by jaydfox - 08/30/2007, 11:20 PM RE: Tetration below 1 - by Daniel - 08/31/2007, 12:47 AM RE: Tetration below 1 - by GFR - 09/02/2007, 01:30 PM RE: Tetration below 1 - by bo198214 - 09/02/2007, 01:40 PM RE: Tetration below 1 - by GFR - 09/02/2007, 05:38 PM RE: Tetration below 1 - by jaydfox - 09/03/2007, 03:58 PM RE: Tetration below 1 - by bo198214 - 09/03/2007, 04:07 PM RE: Tetration below 1 - by jaydfox - 09/03/2007, 04:36 PM RE: Tetration below 1 - by jaydfox - 09/05/2007, 11:24 PM RE: Tetration below 1 - by GFR - 09/06/2007, 12:01 AM RE: Tetration below 1 - by jaydfox - 09/06/2007, 03:28 AM RE: Tetration below 1 - by jaydfox - 09/06/2007, 07:21 AM RE: Tetration below 1 - by bo198214 - 03/26/2008, 04:51 PM

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