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f( f(x) ) = exp(x) solved ! ! !
#21
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
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#22
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 .
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#23
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
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#24
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
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#25
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". Smile

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
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#26
[/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
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