simple idea ... - 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: simple idea ... (/showthread.php?tid=277) simple idea ... - tommy1729 - 04/28/2009 a simple idea ... maybe considered before ? let us approximate exp(x) with polynomials of degree n. let us call that exp_n(x). now we can solve for f_n(f_n(x)) = exp_n(x) if n is large enough and then f_n(x) being a polynomial of degree < n. ( we ignore the terms of degree > n in f_n(f_n(x)) ) now if lim f_n(x) exists we got f(x) with f(f(x) = exp(x). .... headscratch ... regards tommy1729 RE: simple idea ... - andydude - 04/29/2009 tommy1729 Wrote:headscratch ...That doesn't work. Suppose $f(x) = x^2$ then $f(f(x)) = x^4$. Chopping off the second series gives $f(x) = 0$ which is false. RE: simple idea ... - tommy1729 - 04/29/2009 andydude Wrote:tommy1729 Wrote:headscratch ...That doesn't work. Suppose $f(x) = x^2$ then $f(f(x)) = x^4$. Chopping off the second series gives $f(x) = 0$ which is false. well , thanks for your reply andy. but i dont agree , and i assume you misunderstood what i meant. in your example there is nothing to chop off ? the degree of f(f(x)) = 4 just as desired. what is ignored is more something like this ( with some imagination , example not so good ) f(x) = a + b x^4 + (x^17) / 17! f(f(x)) = polynomial of degree 16 + ' some terms of degree > 16 ' ' those terms of degree > 16 ' are caused by the x^17 part so they are dropped. they are pretty small for some x afterall since i took x ^17 / 17! that is important , what we ignore must be relatively small. but exp(x) has a fast converging taylor series so *that* should probably not be a problem. however in the OP i assumed a limit , that might be trickier. but i believe that when taking the limit in a good way , we have chance at succeeding ... i hope this clarifies a bit ... regards tommy1729 RE: simple idea ... - andydude - 04/29/2009 tommy1729 Wrote:i hope this clarifies a bit ... Nope. Still head scratching. RE: simple idea ... - bo198214 - 04/29/2009 andydude Wrote:Nope. Still head scratching. Me too. Most polynomials have no polynomial half iterate, so how can your suggestion work?