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 x exp(x) musing tommy1729 Ultimate Fellow Posts: 1,372 Threads: 336 Joined: Feb 2009 07/01/2009, 03:19 PM let f( x exp(x) ) * exp(x) * (x+1) = f(x) then F( x exp(x) ) = F(x) + C inversing -> G(x+C) = G(x) * exp(G(x)) now notice G(x + n C) = G(x) * exp(G(x)) * exp(G(x))^exp(G(x)) * ... n times. thus exp(G(x)) ^^ n = G(x + n C) / G(x + n C - C) ( Form 1 ) and we have a solution for tetration. now that was very briefly the idea , i know alot relates to it. another related idea is to take the integral ; because sexp'(x) = sexp(x) * sexp(x-1) * sexp(x-2) * ... * C2. so sexp(x) relates to integral G(x). ( " Form 2 " not a formula yet actually :p ) now we have 2 strongly related ideas. and keep in mind that our base needs to be bigger than eta. for real bases > eta ; Form 1 or Form 2 might apply to one of the solutions for tetration ... ( or both sometimes !? since Derivate = prod ) i know , i know , ive only given ideas and equations , not solutions or proofs. but still , i think it is worth consideration. and maybe f(x) can be found by current attempts for tetration. and G(x) might be found by inversing the ( modified ) carleman matrix of F(x) notice also that x exp(x) has a unique real fixed point !! i used 'x' instead of 'z' because mainly we want R > eta -> R btw G(x) should be strictly increasing on the reals. i hope that idea was not retarded regards tommy1729 « Next Oldest | Next Newest »

 Messages In This Thread x exp(x) musing - by tommy1729 - 07/01/2009, 03:19 PM RE: x exp(x) musing - by BenStandeven - 07/03/2009, 03:40 PM RE: x exp(x) musing - by tommy1729 - 07/03/2009, 07:34 PM RE: x exp(x) musing - by BenStandeven - 07/04/2009, 11:25 PM RE: x exp(x) musing - by bo198214 - 07/05/2009, 07:12 PM RE: x exp(x) musing - by tommy1729 - 07/05/2009, 07:33 PM

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