• 1 Vote(s) - 1 Average
• 1
• 2
• 3
• 4
• 5
 Tommy's Gaussian method. tommy1729 Ultimate Fellow Posts: 1,668 Threads: 368 Joined: Feb 2009 07/09/2021, 04:18 AM (This post was last modified: 07/22/2021, 11:55 AM by tommy1729.) Hello everyone ! Time to get serious. Another infinite composition method. This time I took care of unneccessary complications such a branch points, singularities etc.  Periodic points remain a topic however. A sketch of the idea : Let oo denote real infinity. Basically it combines these 3 : 1) https://math.eretrandre.org/tetrationfor...p?tid=1320 2) https://math.eretrandre.org/tetrationfor...p?tid=1326 3) And most importantly the following f(s) ; Tommy's Gaussian method : f(s) = exp(t(s) * f(s-1)) t(s) = (erf(s)+1)/2 Notice that t(s - oo) = 0 and t(s + oo) = 1 for all (finite complex) s. In particular  IF 2 + Re(w)^4 < Re(w)^6 < Im(w)^6 - Re(w)^2  THEN t(w)^2 is close to 0 or 1. Even more so when Im(w)^2 is small compared to Re(w)^2. The continued fraction for t(s) gives a good idea how it grows on the real line ; it grows at speeds about exp(x^2) to 0 or 1. A visual of t(w) would demonstrate that it converges fast to 0 or 1 in the (resp) left and right triangle of an x shaped region. That x shape is almost defined by Re(w)^2 = Im(w)^2 thus making approximately 4 90 degree angles at the origin and having only straith lines. Therefore we can consistantly define for all s without singularities or poles ( hence t(s) and f(s) being entire ! )  f(s) = exp( t(s) * exp( t(s-1) * exp( t(s-2) * ... thereby making f(s) an entire function ! Now we pick a point say e. And we can try the ideas from  1) https://math.eretrandre.org/tetrationfor...p?tid=1320 2) https://math.eretrandre.org/tetrationfor...p?tid=1326 to consistently define  exp^[s](e) and then by analytic continuation from e to z ; exp^[s](z). We know this analytic continuation exists because f(s) is entire and for some appropriate q we must have exp^[q](e) = z. By picking the correct branch we also got the slog function. It should be as simple as ( using small o notation ) lim n to +oo , Re( R ) > 0 ; exp^[R](z) = ln^[n] ( f( g(z) + n + R) ) + o( t(-n+R) ) and ofcourse using the appropriate brances of ln and g. regards tommy1729 Tom Marcel Raes « Next Oldest | Next Newest »

 Messages In This Thread Tommy's Gaussian method. - by tommy1729 - 07/09/2021, 04:18 AM RE: Tommy's Gaussian method. - by JmsNxn - 07/09/2021, 04:56 AM RE: Tommy's Gaussian method. - by JmsNxn - 07/10/2021, 04:34 AM RE: Tommy's Gaussian method. - by JmsNxn - 07/12/2021, 04:48 AM RE: Tommy's Gaussian method. - by tommy1729 - 07/21/2021, 05:29 PM RE: Tommy's Gaussian method. - by tommy1729 - 07/21/2021, 06:55 PM RE: Tommy's Gaussian method. - by tommy1729 - 07/21/2021, 09:52 PM RE: Tommy's Gaussian method. - by JmsNxn - 07/22/2021, 02:21 AM RE: Tommy's Gaussian method. - by tommy1729 - 07/22/2021, 12:13 PM RE: Tommy's Gaussian method. - by JmsNxn - 07/23/2021, 04:13 PM RE: Tommy's Gaussian method. - by tommy1729 - 07/25/2021, 10:54 PM RE: Tommy's Gaussian method. - by JmsNxn - 07/23/2021, 11:18 PM RE: Tommy's Gaussian method. - by tommy1729 - 07/25/2021, 11:20 PM RE: Tommy's Gaussian method. - by tommy1729 - 07/25/2021, 11:58 PM RE: Tommy's Gaussian method. - by JmsNxn - 07/26/2021, 10:24 PM RE: Tommy's Gaussian method. - by JmsNxn - 07/25/2021, 11:59 PM RE: Tommy's Gaussian method. - by tommy1729 - 07/26/2021, 12:03 AM RE: Tommy's Gaussian method. - by tommy1729 - 07/28/2021, 12:02 AM RE: Tommy's Gaussian method. - by JmsNxn - 07/28/2021, 12:24 AM RE: Tommy's Gaussian method. - by tommy1729 - 08/06/2021, 12:15 AM RE: Tommy's Gaussian method. - by tommy1729 - 08/19/2021, 09:40 PM RE: Tommy's Gaussian method. - by tommy1729 - 11/09/2021, 01:12 PM RE: Tommy's Gaussian method. - by tommy1729 - 11/09/2021, 11:59 PM RE: Tommy's Gaussian method. - by tommy1729 - 11/10/2021, 12:10 AM RE: Tommy's Gaussian method. - by JmsNxn - 11/11/2021, 12:58 AM RE: Tommy's Gaussian method. - by tommy1729 - 05/12/2022, 11:58 AM RE: Tommy's Gaussian method. - by tommy1729 - 05/12/2022, 12:01 PM RE: Tommy's Gaussian method. - by tommy1729 - 05/14/2022, 12:25 PM RE: Tommy's Gaussian method. - by tommy1729 - 05/22/2022, 12:35 AM RE: Tommy's Gaussian method. - by JmsNxn - 05/22/2022, 12:40 AM RE: Tommy's Gaussian method. - by tommy1729 - 05/26/2022, 10:54 PM RE: Tommy's Gaussian method. - by JmsNxn - 05/26/2022, 10:57 PM RE: Tommy's Gaussian method. - by tommy1729 - 05/26/2022, 11:06 PM RE: Tommy's Gaussian method. - by JmsNxn - 05/26/2022, 11:13 PM RE: Tommy's Gaussian method. - by tommy1729 - 06/28/2022, 02:23 PM

 Possibly Related Threads… Thread Author Replies Views Last Post semi-group homomorphism and tommy's U-tetration tommy1729 5 65 08/12/2022, 08:14 PM Last Post: tommy1729 Describing the beta method using fractional linear transformations JmsNxn 5 101 08/07/2022, 12:15 PM Last Post: JmsNxn The Etas and Euler Numbers of the 2Sinh Method Catullus 2 135 07/18/2022, 10:01 AM Last Post: Catullus " tommy quaternion " tommy1729 30 8,242 07/04/2022, 10:58 PM Last Post: Catullus tommy's new conjecture/theorem/idea (2022) ?? tommy1729 0 121 06/22/2022, 11:49 PM Last Post: tommy1729 The beta method thesis JmsNxn 9 1,293 04/20/2022, 05:32 AM Last Post: Ember Edison tommy beta method tommy1729 0 601 12/09/2021, 11:48 PM Last Post: tommy1729 The Generalized Gaussian Method (GGM) tommy1729 2 1,319 10/28/2021, 12:07 PM Last Post: tommy1729 Arguments for the beta method not being Kneser's method JmsNxn 54 15,984 10/23/2021, 03:13 AM Last Post: sheldonison tommy's singularity theorem and connection to kneser and gaussian method tommy1729 2 1,251 09/20/2021, 04:29 AM Last Post: JmsNxn

Users browsing this thread: 1 Guest(s)