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 Tetration extension for bases between 1 and eta bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 11/07/2009, 08:12 PM (11/07/2009, 05:11 PM)bo198214 Wrote: But now there is still the question why the limit $f(x)$ indeed satisfies $f(x+1)=b^{f(x)}$? Actually it is not the case but we can obtain something very similar. I just read in [1], p. 31, th. 10, that we have the following limit for a function $f(x)=f_1 x+f_2x^2+f_3 x^3 + ...$ with $0: $\lim_{n\to\infty} \frac{f^{\circ n}(t)-f^{\circ n}(\theta)}{f^{\circ n}(t)-f^{\circ n+1}(t)}=w\frac{1-f_1^w}{1-f_1}$ for $\theta = f^{\circ w}(t)$. If we invert the formula we get $f^{\circ w}(t) = \lim_{n\to\infty} f^{\circ -n}\left(w\frac{1-f_1^w}{1-f_1}\left(f^{\circ n+1}(t)-f^{\circ n}(t)\right)+f^{\circ n}(t)\right)$ In our case though we dont have f(0)=0 but there is some fixed point $z_f$ of $f$, $f(z_f)=z_f$. In this case however the formula is quite similar, the only change is that $f_1=f'(z_f)$. [1] Ecalle: Theorie des invariants holomorphes « 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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