03/17/2009, 11:24 PM

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

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