Truncate where ? tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 03/20/2015, 12:15 AM (This post was last modified: 03/20/2015, 12:19 AM by tommy1729.) Still considering parabolic fixpoints at 0. And also those "semi-Taylor" expansions that have radius 0. For instance solving f(f(x)) = x + x^2. See also : http://math.stackexchange.com/questions/...324#912324 But in general f(f(x)) = x + r x^2 + r_2 x^3 + ... = g(x) (real-entire) with r > 0. Considering the semi-Taylor for f(x) ; Now if we rewrite the logs and sqrt's of x as Taylor series in (x+1) and truncate at A sums and we truncate the remaining Taylor series at A sums , then I think that : f(x) , for every x > 0 can be given by truncating the semi-Taylor at A sums ( as described above ). Actually not A , but A(x) , where A is a function of x. Notice that I did not say A is an integer. So im talking about continuum sums again. ALthough the best fitting integer is also intresting ofcourse. Natural question is ofcourse , for a given g(x) , how to find A(x) ?? It seems easy in a numerical experimental way , so I have hope for this. ( this is the new thread what I talked about in post 5 of http://math.eretrandre.org/tetrationforu...hp?tid=965 ) Experimental math seems easy , considering that the truncation should give values for A(x) such that the truncated f(x) is between x and g(x). symbolic : for Q > x > 0 where Q is the smallest value > 0 where g'(Q) = 0 , x < f_A(x) < g(x). I used f_A(x) for f(x) truncated at the A(x) th term. To give some examples I wonder about : f_A(x) resp A(x) for 1) x + x^N ( some N > 2 ) 2) x exp(x) ... I think A(x) satisfies some logical things. for instance : if A(x) belongs to g(x) , then A(x) + 1 belongs to (g(x)+1) x. Also I think the concept of "growth" (as considered by me and sheldon) is important as are r_2 and r_3. This will probably give nice Visuals. regards tommy1729 « Next Oldest | Next Newest » 