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 tetration limit ?? BenStandeven Junior Fellow Posts: 27 Threads: 3 Joined: Apr 2009 05/01/2009, 03:00 AM BenStandeven Wrote:BenStandeven Wrote:BenStandeven Wrote:Let's see here. The fixed point for base (eta + eps) is e + delta(eps), where delta satisfies: $\delta(\eps) = -e^2 \eps/\eta + \sqrt{2/\eta} e^{3/2} \sqrt{-\eps} + O(\eps^{3/2})$ $(\eta + \eps)^{e + \Re(\delta(\eps)) + \theta} = e^{1 + \theta (1 + e \eps/\eta)/e} + O(\eps^{3/2})$ Now if $\theta$ is on the order of $\sqrt \eps$, we have $(\eta + \eps)^{e + \Re(\delta(\eps)) + \theta} = e + \theta + \theta^2/2e + O(\eps^{3/2})$, so the effect of an additional level of tetration is to add $-\Re(\delta(\eps)) + \theta^2/2e$ to the exponent. To cross this region of length $2 \sqrt \eps$ would require between $2 / (\sqrt \eps (e^2/\eta + 1/2e))$ and $2 / (\sqrt \eps (e^2/\eta))$ steps. But if $\theta$ is of a larger order, the epsilon-dependent terms may be neglected, and we get that $(\eta + \eps)^{e + \theta} = \eta^{e + \theta} + O(\eps)}$. So it takes roughly $slog_{\eta}(e - \sqrt\eps)$ tetration levels to reach $e - \sqrt \eps$. To be continued... Now $slog_{\eta} ( e - \sqrt\eps) = 2e/\sqrt\eps + O(\eps)$. So we see that $\lim_{\eps \to 0} {}^{C/\sqrt\eps}(\eta + \eps)$ is e for any constant from $2e$ to $e^2/\eta$. Also, it is e for any constant less than 2e, since ${}^{C/\sqrt\eps}(\eta) = e - 2e/{C/\sqrt\eps} + O(\eps)$. Similarly, $\lim_{\eps \to 0} {}^{C \eps^{-1/2+\sigma}}(\eta + \eps) = e$ for any positive sigma. But assuming positive sigma again, $\lim_{\eps \to 0} {}^{C \eps^{-1/2-\sigma}}(\eta + \eps)$ would be $\eta + O(\sqrt\eps)$ exponentiated $C \eps^{-1/2-\sigma/2} - D \eps^{-1/2}$ times and then another $C \eps^{-1/2-\sigma} - C \eps^{-1/2-\sigma/2}$; the first operation is enough to move the exponent up to at least $e + C \eps^{-\sigma/2}$, while the next would take it from there to $e + C/2e \eps^{-\sigma}$ and each additional step would square the excess over e again. So we would get at least $e + C' \eps^{-\sigma 2^{C \eps^{-1/2-\sigma} - C \eps^{-1/2-\sigma/2} - 2}}$, which clearly tends to infinity as epsilon approaches zero. So $\lim_{\eps \to 0} {}^{C \eps^{-1/2+\sigma}}(\eta + \eps)$ is e if sigma is 0 or positive (probably independent of C), and infinite if sigma is negative. « Next Oldest | Next Newest »

 Messages In This Thread tetration limit ?? - by tommy1729 - 04/01/2009, 05:49 PM RE: tetration limit ?? - by nuninho1980 - 04/01/2009, 08:15 PM RE: tetration limit ?? - by bo198214 - 04/02/2009, 09:58 PM RE: tetration limit ?? - by nuninho1980 - 04/03/2009, 12:53 AM RE: tetration limit ?? - by bo198214 - 04/03/2009, 12:49 PM RE: tetration limit ?? - by nuninho1980 - 04/03/2009, 05:54 PM RE: tetration limit ?? - by bo198214 - 04/02/2009, 02:50 PM RE: tetration limit ?? - by tommy1729 - 04/02/2009, 09:24 PM RE: tetration limit ?? - by bo198214 - 04/02/2009, 09:56 PM RE: tetration limit ?? - by tommy1729 - 04/02/2009, 10:39 PM RE: tetration limit ?? - by tommy1729 - 05/29/2011, 07:28 PM RE: tetration limit ?? - by bo198214 - 05/31/2011, 10:34 AM RE: tetration limit ?? - by nuninho1980 - 04/03/2009, 06:12 PM RE: tetration limit ?? - by bo198214 - 04/06/2009, 10:49 PM RE: tetration limit ?? - by nuninho1980 - 04/07/2009, 01:35 AM Updated results for tetration limit - by sheldonison - 10/31/2010, 03:32 PM RE: tetration limit ?? - by nuninho1980 - 04/04/2009, 02:21 PM RE: tetration limit ?? - by gent999 - 04/14/2009, 10:12 PM RE: tetration limit ?? - by bo198214 - 04/14/2009, 10:31 PM RE: tetration limit ?? - by gent999 - 04/15/2009, 12:18 AM RE: tetration limit ?? - by bo198214 - 04/15/2009, 01:35 PM RE: tetration limit ?? - by tommy1729 - 04/15/2009, 03:05 PM RE: tetration limit ?? - by gent999 - 04/15/2009, 04:41 PM RE: tetration limit ?? - by tommy1729 - 04/29/2009, 01:08 PM RE: tetration limit ?? - by BenStandeven - 04/30/2009, 11:29 PM RE: tetration limit ?? - by tommy1729 - 04/30/2009, 11:38 PM RE: tetration limit ?? - by BenStandeven - 05/01/2009, 01:35 AM RE: tetration limit ?? - by BenStandeven - 05/01/2009, 01:00 AM RE: tetration limit ?? - by JmsNxn - 04/14/2011, 08:17 PM RE: tetration limit ?? - by tommy1729 - 05/28/2011, 12:28 PM RE: tetration limit ?? - by nuninho1980 - 10/31/2010, 10:31 PM RE: tetration limit ?? - by JmsNxn - 05/29/2011, 02:06 AM RE: tetration limit ?? - by tommy1729 - 05/14/2015, 08:29 PM RE: tetration limit ?? - by tommy1729 - 05/14/2015, 08:33 PM RE: tetration limit ?? - by tommy1729 - 05/28/2015, 11:32 PM RE: tetration limit ?? - by sheldonison - 06/11/2015, 10:27 AM RE: tetration limit ?? - by sheldonison - 06/15/2015, 01:00 AM RE: tetration limit ?? - by tommy1729 - 06/01/2015, 02:04 AM RE: tetration limit ?? - by tommy1729 - 06/11/2015, 08:25 AM

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