• 1 Vote(s) - 5 Average
• 1
• 2
• 3
• 4
• 5
 tetration limit ?? BenStandeven Junior Fellow Posts: 27 Threads: 3 Joined: Apr 2009 05/01/2009, 01:00 AM (This post was last modified: 05/01/2009, 01:04 AM by BenStandeven.) BenStandeven Wrote:Let's see here. The fixed point for base (eta + eps) is e + delta(eps), where delta satisfies: $\delta(\eps) = -e^2 \eps/\eta + \sqrt{2/\eta} e^{3/2} \sqrt{-\eps} + O(\eps^{3/2})$ Now $(\eta + \eps)^{e + \Re(\delta(\eps))} = (\eta + \eps)^{e + -e^2 \eps/\eta + O(\eps^{3/2})} = \e( \ln(\eta + \eps) (e + -e^2 \eps/\eta + O(\eps^{3/2})))$, which is: $\e( 1 + -e \eps/\eta + \ln(1 + \eps/\eta) (e + -e^2 \eps/\eta) + O(\eps^{3/2})) = \e( 1 + O(\eps^{3/2})) = e + O(\eps^{3/2})$. So: $(\eta + \eps)^{e + \Re(\delta(\eps)) + \theta} = (\eta + \eps)^{(e + \Re(\delta(\eps)))(1 + \theta/(e + \Re(\delta(\eps))))} = e^{(1 + O(\eps^{3/2}))(1 + \theta (1 - \Re(\delta(\eps))/e + O(\eps^2))/ e )}$ which for $|\theta| << 1/\sqrt{\eps}$ is $e^{1 + \theta (1 + e \eps/\eta)/ e + O(\eps^{3/2}) } = e^{1 + \theta (1 + e \eps/\eta)/e} + O(\eps^{3/2})$ To be continued... « 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

 Possibly Related Threads... Thread Author Replies Views Last Post Generalized Kneser superfunction trick (the iterated limit definition) MphLee 25 9,178 05/26/2021, 11:55 PM Last Post: MphLee Dangerous limits ... Tommy's limit paradox tommy1729 0 3,587 11/27/2015, 12:36 AM Last Post: tommy1729 Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x Ivars 10 23,420 03/29/2015, 08:02 PM Last Post: tommy1729 Another limit tommy1729 0 3,088 03/18/2015, 06:55 PM Last Post: tommy1729 A limit exercise with Ei and slog. tommy1729 0 3,502 09/09/2014, 08:00 PM Last Post: tommy1729 [MSE] The mick tommy limit conjecture. tommy1729 1 4,762 03/30/2014, 11:22 PM Last Post: tommy1729 tetration base conversion, and sexp/slog limit equations sheldonison 44 95,366 02/27/2013, 07:05 PM Last Post: sheldonison Solve this limit Nasser 4 8,662 12/03/2012, 07:46 AM Last Post: Nasser (MSE) A limit- question concerning base-change Gottfried 0 4,007 10/03/2012, 06:44 PM Last Post: Gottfried a limit curiosity ? Pi/2 tommy1729 0 3,556 08/07/2012, 09:27 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)