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f(f(x)) = exp(x) - x

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master1729

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Nov 30, 2009, 5:43:05 PM11/30/09
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let f(x) be Coo.

f(f(x)) = exp(x) - x

what is f(x) ?

does the (only) fixpoint at - oo help ?

can f(x) be entire ?

is there a solution f(x) such that f(x) has no fixpoint apart from - oo ?

can f(x) be expressed in terms of tetration ?

is there a solution f(f(x)) = exp(x) - x with f(x) E Coo , f(x) mapping all reals to reals and f(x) having no fixpoint apart from - oo ?

does " a solution f(f(x)) = exp(x) - x with f(x) E Coo , f(x) mapping all reals to reals and f(x) having no fixpoint apart from - oo " imply that all derivatives are strictly positive reals ?

can f(x) be expressed in terms of pentation ?

does the substitution y = 1/x help ?
( trying to 'move the fixpoint' but problems occur e.g. exp(1/x) has a singularity at 0 ! )

this seems like a difficult problem ...

regards

tommy1729

master1729

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Dec 1, 2009, 3:08:47 PM12/1/09
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stop looking at monty hall and inaccessible cardinals and take a look at this.

surely questions of tommy1729 must be trivial hmm
this is your glorious moment to show my questions are trivial and/or you are brilliant.

master1729

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Dec 2, 2009, 11:47:02 AM12/2/09
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AAARGGHH

I MEANT f(f(x)) = exp(x) + x !!

tommy1729

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