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 Tetration extension for bases between 1 and eta bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 12/15/2009, 01:40 AM (This post was last modified: 12/15/2009, 01:40 AM by bo198214.) (12/15/2009, 01:01 AM)dantheman163 Wrote: Upon closer study i think i have found a formula that actually works. ${}^ x b = \lim_{k\to \infty} \log_b ^k({}^ k b (\ln(b){}^ \infty b)^x-{}^ \infty b(\ln(b){}^ \infty b)^x+{}^ \infty b)$ Also i have noticed that this can be more generalized to say, if $f(x)=b^x$ then $f^n(x)= \lim_{k\to \infty} \log_b ^k( \exp_b^k(x) (\ln(b){}^ \infty b)^n-{}^ \infty b(\ln(b){}^ \infty b)^n+{}^ \infty b)$ Actually you rediscovered the Kœnigs formula (2.24 in the (unfinished) overview paper). $f^{[w]}(z)=\lim_{k\to\infty} f^{[-k]}\left((1-\lambda^{w})\cdot p+\lambda^{w}\cdot f^{[k]}(z)\right)$ for $f(x)=b^x$. $\lambda$ is the derivative at the fixed point $p={^\infty b}$, which is $\lambda=\ln(p)=\ln(b^p)=p\ln(b)=\ln(b){^\infty b}$. Good work! « 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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