• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Possible continuous extension of tetration to the reals Dasedes Newbie Posts: 1 Threads: 1 Joined: Oct 2016 10/10/2016, 04:57 AM I've been sort of obsessed with Tetratipn since I first heard about it, and I've scoured the internet trying to learn all that I could about it. I came across this oeis Talk: Iterated Logarithms recently and the idea of a weighted function to deal with non-Natural numbers struck me as interesting. I've played with his idea and came up with this: $^ya = {a_{_0}}^.^{.}^.^{(a_{h-1} - ((a - 1) - r))}$ and $log_b*(^ya) = \lceil slog_b(^ya) \rceil = \lceil y \rceil$ $slog_b(^ya) = log_b*(^ya) - r$ Where $r = y - \lfloor y \rfloor$ and $h = \lfloor y + 1 \rfloor$ This leads to a continuous function that looks like a very steep exponential graph. It also leads to several identities: $slog_b(^ya) = yslog_b(a)$ $^0a = a_0 - (a - 1) = a - a + 1 = 1;$$h = \lfloor 0 + 1 \rfloor = 1$ $^{-y}a = \frac {1}{a_{_0}^.^{.}^.^{(a_{h-1} - ((a - 1) - r))}}$ Please tell me your thoughts on this, thanks. ~Dasedes « Next Oldest | Next Newest »

 Possibly Related Threads… Thread Author Replies Views Last Post On extension to "other" iteration roots Leo.W 31 2,392 Yesterday, 06:45 AM Last Post: JmsNxn Continuous Hyper Bouncing Factorial Catullus 9 120 08/15/2022, 07:54 AM Last Post: JmsNxn Qs on extension of continuous iterations from analytic functs to non-analytic Leo.W 17 825 08/10/2022, 11:34 PM Last Post: JmsNxn Unifying continuous and discrete physics Daniel 0 39 07/31/2022, 01:26 PM Last Post: Daniel Tetration extension for bases between 1 and eta dantheman163 23 35,224 07/05/2022, 04:10 PM Last Post: Leo.W Non-trivial extension of max(n,1)-1 to the reals and its iteration. MphLee 9 8,213 06/15/2022, 10:59 PM Last Post: MphLee Ueda - Extension of tetration to real and complex heights MphLee 4 1,244 05/08/2022, 11:48 PM Last Post: JmsNxn Andrew Robbins' Tetration Extension bo198214 32 77,445 08/22/2016, 04:19 PM Last Post: Gottfried [Update] Comparision of 5 methods of interpolation to continuous tetration Gottfried 30 57,981 02/04/2014, 12:31 AM Last Post: Gottfried [UFO] - a contradiction in assuming continuous tetration? Gottfried 18 37,713 08/29/2010, 08:44 PM Last Post: Gottfried

Users browsing this thread: 1 Guest(s)