• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Another asymptotic development, similar to 2sinh method JmsNxn Ultimate Fellow Posts: 902 Threads: 111 Joined: Dec 2010 07/05/2011, 06:34 PM (This post was last modified: 07/05/2011, 07:37 PM by JmsNxn.) http://math.eretrandre.org/tetrationforu...hp?tid=635 Thanks to mike's post here (near the bottom), we are given an asymptotic development for $e^x \cdot \ln(x)$, namely: $e^x \cdot \ln(x) \sim \sum_{n=0}^{\infty} x^n \frac{\psi_0(n+1)}{n!}$ which after solving for ln(x) gives: $\ln(x) \sim \sum_{n=0}^{\infty} x^n (\sum_{k=0}^{n} (-1)^k \cdot \frac{\psi_0(n-k)}{k!(n-k)!})$ I'm unsure about the convergence radius of this series, I've been unable to prove anything about it, however I'll continue with the conjecture leaving it open whether the series converges infinitely or not. It's a product of two infinite convergent series, so I hope it converges. Given this development, we can write: $\lambda(x) = \sum_{n=0}^{\infty} x^n (\sum_{k=0}^{n} (-1)^k \cdot \frac{\psi_0(n-k)}{k!(n-k)!})$ which I'm wondering if it's possible to perform regular iteration on lambda. If it is, we are given a formula similar to Tommy's 2sinh method. $\exp^{\circ z}(x) = \lim_{n\to\infty} \ln^{\circ n}(\lambda^{\circ -z}(\exp^{\circ n}(x)))$ I'm just posting this because it popped in my head and I thought it would be interesting. Could this method actually work (for real x, that is)? I'm unsure about performing regular iteration on lambda. I couldn't find a fixpoint :\, so maybe it's not possible. For bases other than e we are given: $\lambda_b(x) = \frac{\lambda(x)}{\ln(b)} = \sum_{n=0}^{\infty} \frac{x^n}{\ln(b)} (\sum_{k=0}^{n} (-1)^k \cdot \frac{\psi_0(n-k)}{k!(n-k)!})$ $\exp_b^{\circ z}(x) = \lim_{n\to\infty} \log_b^{\circ n}(\lambda_b^{\circ -z}(\exp_b^{\circ n}(x)))$ I think actually for other bases we're more likely to find a fixpoint, so regular iteration might be easier. For base eta the fixpoint occurs at around 1.9 « Next Oldest | Next Newest »

 Possibly Related Threads… Thread Author Replies Views Last Post Searching for an asymptotic to exp[0.5] tommy1729 203 378,060 08/07/2022, 10:42 PM Last Post: tommy1729 Describing the beta method using fractional linear transformations JmsNxn 5 87 08/07/2022, 12:15 PM Last Post: JmsNxn The Etas and Euler Numbers of the 2Sinh Method Catullus 2 134 07/18/2022, 10:01 AM Last Post: Catullus A Limit Involving 2sinh Catullus 0 83 07/17/2022, 06:15 AM Last Post: Catullus Tetration Asymptotic Series Catullus 18 892 07/05/2022, 01:29 AM Last Post: JmsNxn Tommy's Gaussian method. tommy1729 34 9,221 06/28/2022, 02:23 PM Last Post: tommy1729 Reviving an old idea with 2sinh. tommy1729 7 369 06/28/2022, 02:14 PM Last Post: tommy1729 The beta method thesis JmsNxn 9 1,288 04/20/2022, 05:32 AM Last Post: Ember Edison tommy beta method tommy1729 0 595 12/09/2021, 11:48 PM Last Post: tommy1729 The Generalized Gaussian Method (GGM) tommy1729 2 1,314 10/28/2021, 12:07 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)