• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Tensor power series andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 05/22/2008, 04:36 AM (This post was last modified: 05/22/2008, 04:39 AM by andydude.) I just realized I never explained how that function is related to the exponential factorial. The exponential factorial is defined as $EF(x) = x^{EF(x-1)}$ where $EF(1) = 1$, which is a univariate function. To make this function from the function above, you can first notice the pattern in its iterates: $ \begin{tabular}{rl} E^{\circ 1}(c, 1) = E(c, 1) & = (1, 2) \\ E^{\circ 2}(c, 1) = E(1, 2) & = (2^1, 3) \\ E^{\circ 3}(c, 1) = E(2, 3) & = (3^{2^1}, 4) \\ E^{\circ 4}(c, 1) = E(9, 4) & = (4^{3^{2^1}}, 5) \end{tabular}$ where c could theoretically be anything, since $1^c = 1$. And since I haven't indicated whether we are using 0-based or 1-based vectors, I'm going to use another contraction to get the top value: $ EF(x) = \left[{1 \atop 0}\right] \cd E^{\circ x}\left[{c \atop 1}\right]$ Now it should be obvious from this that $EF(0) = c$ which begs the question in my Conjectures post: what is EF(0)? Andrew Robbins « Next Oldest | Next Newest »

 Messages In This Thread Tensor power series - by andydude - 05/13/2008, 07:58 AM RE: Tensor power series - by andydude - 05/13/2008, 07:59 AM RE: Tensor power series - by andydude - 05/13/2008, 08:11 AM RE: Tensor power series - by andydude - 05/14/2008, 06:18 AM RE: Tensor power series - by Gottfried - 05/20/2008, 08:39 PM RE: Tensor power series - by andydude - 05/22/2008, 12:58 AM RE: Tensor power series - by andydude - 05/22/2008, 04:11 AM RE: Tensor power series - by andydude - 05/22/2008, 04:36 AM RE: Tensor power series - by bo198214 - 05/24/2008, 10:10 AM RE: Tensor power series - by andydude - 06/04/2008, 08:08 AM

 Possibly Related Threads... Thread Author Replies Views Last Post Perhaps a new series for log^0.5(x) Gottfried 3 394 03/21/2020, 08:28 AM Last Post: Daniel A Notation Question (raising the highest value in pow-tower to a different power) Micah 8 3,293 02/18/2019, 10:34 PM Last Post: Micah Taylor series of i[x] Xorter 12 12,642 02/20/2018, 09:55 PM Last Post: Xorter Functional power Xorter 0 1,423 03/11/2017, 10:22 AM Last Post: Xorter 2 fixpoints related by power ? tommy1729 0 1,572 12/07/2016, 01:29 PM Last Post: tommy1729 Taylor series of cheta Xorter 13 13,708 08/28/2016, 08:52 PM Last Post: sheldonison Inverse power tower functions tommy1729 0 1,961 01/04/2016, 12:03 PM Last Post: tommy1729 Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 5,421 05/06/2014, 09:47 PM Last Post: tommy1729 [integral] How to integrate a fourier series ? tommy1729 1 2,669 05/04/2014, 03:19 PM Last Post: tommy1729 about power towers and base change tommy1729 7 8,716 05/04/2014, 08:30 AM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)