• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 base holomorphic tetration mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 11/06/2009, 11:58 AM (This post was last modified: 11/06/2009, 12:11 PM by mike3.) Also, fidding around with this some more, it seems that we cannot continue below $e^{-e}$ in one step, suggesting that it is a singularity (likely a branchpoint).Trying to continue into the pseudocircle with this method from some direction in the complex plane also does not seem to work well, at least with only one step, suggesting lots of singularities/branchpoints in this area. This is Weird!!! Yet we can go up the real axis past $e^{1/e}$... It seems possible that certain Hopfian bifurcation points (of the integer tetrations) are also branchpoints, but not all (note that if one looks at the tetration fractal, the point $b = e^{1/e}$ is an accumulation point of Hopfian points, yet if they were all singularities it would be impossible to analytically continue to $b \g e^{1/e}$, yet this seems to work OK.). A graph might be useful in figuring out what's going on here, but solving the regular iteration near the edges of the Shell-Thron region is difficult, esp. if we want a high-resolution plot. Perhaps if there was some way to evaluate the regular iteration on the edge of the region? But this is tough -- see the thread about regularly iterating $b = e^{-e}$... Note we can parameterize the edge using exponentials, however (see here: http://math.eretrandre.org/tetrationforu...11#pid3011), so obtaining points on it is not a problem. All these wild things make me think that if tetration is ever developed to the point where it could be a "real" special function like the gamma function, error function, zeta function, hypergeometric, Lambert W, etc. it'd be one of the most exotic yet. Especially considering the "fractal" structure (looks like the dynamical Julia set of the exp. function!) it produces for bases like those greater than $e^{1/e}$ when it is plotted for complex values of the tower. « Next Oldest | Next Newest »

 Messages In This Thread base holomorphic tetration - by bo198214 - 11/05/2009, 02:12 PM RE: base holomorphic tetration - by Base-Acid Tetration - 11/05/2009, 10:50 PM RE: base holomorphic tetration - by mike3 - 11/06/2009, 04:15 AM RE: base holomorphic tetration - by mike3 - 11/06/2009, 11:58 AM RE: base holomorphic tetration - by bo198214 - 11/06/2009, 12:12 PM RE: base holomorphic tetration - by mike3 - 11/06/2009, 09:16 PM RE: base holomorphic tetration - by bo198214 - 11/06/2009, 11:29 PM RE: base holomorphic tetration - by mike3 - 11/07/2009, 12:23 AM RE: base holomorphic tetration - by bo198214 - 11/07/2009, 08:17 AM RE: base holomorphic tetration - by mike3 - 11/07/2009, 08:21 AM RE: base holomorphic tetration - by bo198214 - 11/07/2009, 09:55 AM RE: base holomorphic tetration - by bo198214 - 11/07/2009, 04:47 PM RE: base holomorphic tetration - by bo198214 - 11/08/2009, 05:39 PM RE: base holomorphic tetration - by mike3 - 11/08/2009, 08:27 PM RE: base holomorphic tetration - by mike3 - 11/08/2009, 08:25 PM RE: base holomorphic tetration - by bo198214 - 11/08/2009, 08:44 PM RE: base holomorphic tetration - by mike3 - 11/08/2009, 09:51 PM

 Possibly Related Threads... Thread Author Replies Views Last Post Complex Tetration, to base exp(1/e) Ember Edison 7 2,977 08/14/2019, 09:15 AM Last Post: sheldonison Can we get the holomorphic super-root and super-logarithm function? Ember Edison 10 4,726 06/10/2019, 04:29 AM Last Post: Ember Edison Is bounded tetration is analytic in the base argument? JmsNxn 0 1,553 01/02/2017, 06:38 AM Last Post: JmsNxn holomorphic binary operators over naturals; generalized hyper operators JmsNxn 15 17,847 08/22/2016, 12:19 AM Last Post: JmsNxn tetration base sqrt(e) tommy1729 2 3,657 02/14/2015, 12:36 AM Last Post: tommy1729 Explicit formula for the tetration to base [tex]e^{1/e}[/tex]? mike3 1 3,257 02/13/2015, 02:26 PM Last Post: Gottfried tetration base > exp(2/5) tommy1729 2 3,430 02/11/2015, 12:29 AM Last Post: tommy1729 regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 Gottfried 7 10,088 06/25/2013, 01:37 PM Last Post: sheldonison tetration base conversion, and sexp/slog limit equations sheldonison 44 60,063 02/27/2013, 07:05 PM Last Post: sheldonison simple base conversion formula for tetration JmsNxn 0 3,481 09/22/2011, 07:41 PM Last Post: JmsNxn

Users browsing this thread: 1 Guest(s)