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 Brute force tetration A_k(s) is analytic ! tommy1729 Ultimate Fellow Posts: 1,471 Threads: 352 Joined: Feb 2009 03/18/2021, 01:08 AM (This post was last modified: 03/18/2021, 01:11 AM by tommy1729.) I propose an analytic tetration here. One very similar to other ideas from this year. Imo very natural and just requiring alot of simple computions. I dare say brute force. Consider for real k > 5 and Re(s) > 0 : $A_k(s)=exp(A_k(s-1) - exp(-ks))$ We can solve this by infinite composition : $A_k(s)=exp( - exp(-ks) + exp( - exp(-k(s-1)) + ...)$ ( yes to be formal we need a " z parameter " too, but that is a detail ) Now consider the functions $A_5(s),A_6(s),A_7(s),...$ The limit k to +oo gives us A_oo(s) = A(s). Now A(s) satisfies for Re(s) > 0 : A(0) = exp( -exp(-oo) + exp(-oo + 0)) = exp(0 + exp(-oo)) = exp(0) = 1. and  A(s+1) = exp(A(s) + 0) = exp(A(s)) So assuming the limit exists this is analytic tetration on the halfplane Re(s) > 1. By analytic continuation ( taking log's ) this is analytic tetration !! In fact I think it has no singularities for Re(s) > 1 making it perhaps satisfy a uniqueness condition ?? The limit is designed to ( when it exists ) grow slowly on the complex plane (in the imaginary direction ) for Re(s) > 1 therefore probably remaining analytic. We could in principle even approximate this in say excel. regards Tom Marcel Raes tommy1729 « 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

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