• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Brute force tetration A_k(s) is analytic ! JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 03/22/2021, 11:39 PM (This post was last modified: 03/22/2021, 11:50 PM by JmsNxn.) Another way I was thinking, that still has the periodic problem, but is more natural because asymptotically it looks like tetration. $ \Phi(s) = \Omega_{j=1}^\infty \frac{e^z}{e^{j-s} + 1}\bullet z\\$ Where, $ \Phi(s+1) = \frac{e^{\Phi(s)}}{e^{-s}+ 1}\\$ And asymptotically, $ \Phi(s+1)/e^{\Phi(s)} \to 1\\$ So that logarithms may behave better in the complex plane here. I wouldn't worry too much about periodicity because we can always think of the logs collecting $2 \pi i$'s. This way might actually work with holomorphy. Can't be sure though. It definitely constructs $\mathcal{C}^\infty$ tetration on $(-2,\infty)$. But now, we have to deal with poles when $\Im(s) = \pi i$. With this we should expect, $ \log \Phi(s+1) = \Phi(s) + \mathcal{O}(e^{-s})\\$ which looks ripe for convergence in the complex plane. « Next Oldest | Next Newest »

 Messages In This Thread Brute force tetration A_k(s) is analytic ! - by tommy1729 - 03/18/2021, 01:08 AM RE: Brute force tetration A_k(s) is analytic ! - by tommy1729 - 03/18/2021, 12:55 PM RE: Brute force tetration A_k(s) is analytic ! - by JmsNxn - 03/18/2021, 09:11 PM RE: Brute force tetration A_k(s) is analytic ! - by tommy1729 - 03/19/2021, 01:04 PM RE: Brute force tetration A_k(s) is analytic ! - by tommy1729 - 03/19/2021, 01:22 PM RE: Brute force tetration A_k(s) is analytic ! - by Gottfried - 03/19/2021, 05:22 PM RE: Brute force tetration A_k(s) is analytic ! - by JmsNxn - 03/19/2021, 09:12 PM RE: Brute force tetration A_k(s) is analytic ! - by tommy1729 - 03/20/2021, 12:34 AM RE: Brute force tetration A_k(s) is analytic ! - by JmsNxn - 03/22/2021, 08:58 PM RE: Brute force tetration A_k(s) is analytic ! - by JmsNxn - 03/22/2021, 11:39 PM

 Possibly Related Threads... Thread Author Replies Views Last Post Nixon-Banach-Lambert-Raes tetration is analytic , simple and “ closed form “ !! tommy1729 11 4,389 02/04/2021, 03:47 AM Last Post: JmsNxn [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 3,100 03/20/2018, 12:16 AM Last Post: tommy1729 Analytic matrices and the base units Xorter 2 5,516 07/19/2017, 10:34 AM Last Post: Xorter Non-analytic Xorter 0 3,098 04/04/2017, 10:38 PM Last Post: Xorter A conjectured uniqueness criteria for analytic tetration Vladimir Reshetnikov 13 22,861 02/17/2017, 05:21 AM Last Post: JmsNxn Is bounded tetration is analytic in the base argument? JmsNxn 0 3,133 01/02/2017, 06:38 AM Last Post: JmsNxn Are tetrations fixed points analytic? JmsNxn 2 6,440 12/14/2016, 08:50 PM Last Post: JmsNxn The bounded analytic semiHyper-operators JmsNxn 2 7,008 05/27/2016, 04:03 AM Last Post: JmsNxn Periodic analytic iterations by Riemann mapping tommy1729 1 4,671 03/05/2016, 10:07 PM Last Post: tommy1729 Bounded Analytic Hyper operators JmsNxn 25 40,018 04/01/2015, 06:09 PM Last Post: MphLee

Users browsing this thread: 1 Guest(s)