 a vague idea for f(f(x)) = exp(x) - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- 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