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+--- Thread: a vague idea for f(f(x)) = exp(x) (/showthread.php?tid=255)
a vague idea for f(f(x)) = exp(x) - tommy1729 - 03/17/2009
a vague idea ...
f(f(x)) = exp(x)
=> f(f(x)) + O(x) * f(x) = exp(x) with O(x) approaching 0.
=> f_n-1(f_n-1(x)) - exp(x) = - f_n(x)/ n!
=> - n! * ( f_n-1(f_n-1(x)) - exp(x) ) = f_n(x)
so starting at a suitable f_0(x) , whatever it may be , and applying the iteration :
- n! * ( f_n-1(f_n-1(x)) - exp(x) ) = f_n(x)
until it converges ( lim n -> oo ) at " f_oo(x) "
( convergeance depends upon f_0(x) of course )
if f_oo(x) is analytic ,
we have found f_oo(x) = f(x) with f(f(x)) =exp(x)
!!!!!!!!
i know : a vague idea.
but perhaps promising.
regards
tommy1729