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 f( f(x) ) = exp(x) solved ! ! ! tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 02/11/2009, 05:32 PM bo198214 Wrote:Existence was shown by Kneser. Uniqueness is a mostly unresearched yet. Sure is that it is not unique by analyticity alone. well , i assume the existence of tetration ( knesers exp(F(x)) = F(x+1)) leads to existence of f(f(x)) = exp(x). however , im not sure that f(f(x)) = exp(x) is not unique by analyticity alone ? care to explain ? regards tommy1729 bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 02/11/2009, 06:04 PM tommy1729 Wrote:well , i assume the existence of tetration ( knesers exp(F(x)) = F(x+1)) leads to existence of f(f(x)) = exp(x). Yes, as we assume to be strictly increasing we can take the inverse function . Then you can verify that satisfies Quote:however , im not sure that f(f(x)) = exp(x) is not unique by analyticity alone ? care to explain ? Now if we have one analytic solution then we have a lot of other analytic solutions given by for any 1-periodic analytic function (prove that too!) If we make the amplitude of those sufficiently small, then is strictly increasing and is too. Finally: is another analytic solution of . tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 02/11/2009, 08:05 PM bo198214 Wrote:Quote:however , im not sure that f(f(x)) = exp(x) is not unique by analyticity alone ? care to explain ? Now if we have one analytic solution then we have a lot of other analytic solutions given by for any 1-periodic analytic function (prove that too!) If we make the amplitude of those sufficiently small, then is strictly increasing and is too. Finally: is another analytic solution of . i had expected such a reply. but i disagree. let F( real ) map to reals. and let f( real ) map to reals. assuming those are satisfied , i feel that F(x+1) = exp(F(x)) should also satisfy F(x+1/2) = f(F(x)) where f(f(x)) = exp(x) , if it wants to be tetration. in general F(x+a) = f_a(F(x)) where f_a satisfies f_a(((... a times ...(x)))) = exp(x) should be satisfied. furthermore the inverse of F might be multivalued !! but that doesnt mean f(x) is all the possible results of F(1/2 + invF(x)) taking that into account , its probably clear that i will only accept examples of 2 distinct analytic solutions f(x) that map all reals to a subset of reals and satisfy f(f(x)) = exp(x). im not trying to be difficult. but this is important. working with F(x) seems like overkill , instead i focus on f : f(f(x)) = exp(x). ( as an analogue : there are multiple functions that satisfy f(x+1) = e*f(x) but that doesnt mean there are multiple functions that are a solution to exp( log(x) + 1 ) ( being e*x ) ) regards tommy1729 bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 02/11/2009, 10:56 PM I dont really see what you mean. tommy1729 Wrote:taking that into account , its probably clear that i will only accept examples of 2 distinct analytic solutions f(x) that map all reals to a subset of reals and satisfy f(f(x)) = exp(x). And I gave you exactly those two distinct analytic functions: and Quote:( as an analogue : there are multiple functions that satisfy f(x+1) = e*f(x) but that doesnt mean there are multiple functions that are a solution to exp( log(x) + 1 ) ( being e*x ) ) Let one solution be and let the other solution be some other solution be . Then but andydude Long Time Fellow    Posts: 509 Threads: 44 Joined: Aug 2007 02/14/2009, 04:18 AM bo198214 Wrote:the equation was not yet really solved, though we collected several approaches on that question on the forum.Agreed. I should have said "has several approaches" instead of "solved". bo198214 Wrote:What can be solved by regular iteration is , though there are convergence issues, I think this is what you refer to. Well, because and are topologically conjugate (where ), I kind of skipped a step and thought of F (which has a fixed point of 0) and G (which has a fixed point of w) at the same time. Also, I do understand the general idea of tommy's method, to make a function with a fixed point at 0, as a limit of a function with a fixed point not at zero. It makes sense. I just need to sit down at look over the math for a week or so, to convince myself that it all works, and that it gives new insight to the problem. Andrew Robbins tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 02/17/2009, 12:30 AM [/quote] Well, because and are topologically conjugate (where ), I kind of skipped a step and thought of F (which has a fixed point of 0) and G (which has a fixed point of w) at the same time. Also, I do understand the general idea of tommy's method, to make a function with a fixed point at 0, as a limit of a function with a fixed point not at zero. It makes sense. I just need to sit down at look over the math for a week or so, to convince myself that it all works, and that it gives new insight to the problem. Andrew Robbins [/quote] ive been absent for a few days , but im thinking ( working ? ) on related ideas. such as convergeance acceleration. or a zero at other points then 0. i dont think my fixed point at 0 gives a convex solution to tetration ... i will post a conjecture soon. regards tommy1729 « Next Oldest | Next Newest »

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