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 Multiple exp^[1/2](z) by same sexp ? tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 04/29/2014, 12:19 PM Consider the analytic solution sexp(z) that has the minimal amount of singularities => only singularities on the negative real axis. ( uniqueness has been shown , related is TPID 4 : http://math.eretrandre.org/tetrationforu...hp?tid=747) Now let sexp(A) = u where Re(A) > 0,Im(A) > 0 and exp^[v](u) = u for real positive v. Notice that most u have such a v because of the chaotic behaviour of the iterations of exp. ( we ignore branches and functional equations in this sentense ) Now if such A,u are defined on the fundamental branch of sexp such that we have sexp(z+1)=exp(sexp(z)) then we have to conclude something special : exp^[v](u)= u = sexp(A) = sexp(A+v) Now if all if for w and v : 0 < w < 1 << v exp^w(u) = sexp(A + w) = sexp(A + v + w) = sexp(A+ 2v + w) THEN WE MUST CONCLUDE BY INDUCTION AND ANALYTIC CONTINUATION : -------------------------------------- sexp is periodic with period v !! -------------------------------------- But this contradicts the nonperiodicity of sexp and the branches of sexp !! Hence we must have for some u : set w = 1/2 then sexp(A+1/2 + v) =/= sexp(A+1/2) THUS ----------------------------------------------------------------------------- exp^[1/2](u) IS NOT DEFINED UNIQUELY BY THE SAME BRANCH OF SEXP THAT INCLUDES A ----------------------------------------------------------------------------- EUREKA ! this is surprising not ? this reminds me of the recent threads such as "slog(sexp(z)" AND MUCH MORE of a thread by gottfried : http://math.eretrandre.org/tetrationforu...hp?tid=499 I bet you can see the relevance. regards tommy1729 « Next Oldest | Next Newest »

 Messages In This Thread Multiple exp^[1/2](z) by same sexp ? - by tommy1729 - 04/29/2014, 12:19 PM RE: Multiple exp^[1/2](z) by same sexp ? - by sheldonison - 04/29/2014, 07:02 PM RE: Multiple exp^[1/2](z) by same sexp ? - by tommy1729 - 04/29/2014, 10:04 PM RE: Multiple exp^[1/2](z) by same sexp ? - by sheldonison - 04/30/2014, 10:47 AM RE: Multiple exp^[1/2](z) by same sexp ? - by tommy1729 - 04/30/2014, 12:39 PM RE: Multiple exp^[1/2](z) by same sexp ? - by sheldonison - 04/30/2014, 01:08 PM RE: Multiple exp^[1/2](z) by same sexp ? - by tommy1729 - 04/30/2014, 09:16 PM RE: Multiple exp^[1/2](z) by same sexp ? - by tommy1729 - 05/05/2014, 12:25 PM RE: Multiple exp^[1/2](z) by same sexp ? - by sheldonison - 05/06/2014, 09:14 AM RE: Multiple exp^[1/2](z) by same sexp ? - by tommy1729 - 05/06/2014, 09:12 PM RE: Multiple exp^[1/2](z) by same sexp ? - by tommy1729 - 05/05/2014, 10:13 PM RE: Multiple exp^[1/2](z) by same sexp ? - by tommy1729 - 05/06/2014, 09:59 PM RE: Multiple exp^[1/2](z) by same sexp ? - by tommy1729 - 05/06/2014, 10:55 PM

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