• 1 Vote(s) - 5 Average
• 1
• 2
• 3
• 4
• 5
 2 fixpoints , 1 period --> method of iteration series tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 12/21/2016, 01:27 PM Some credit Goes to Gottfried. Assume a strictly increasing analytic function f(x) with fixpoints A and B such that B > A > 0 and F(x) > 0 for x > 0 And also the Period of the first fixpoint = the Period of the second fixpoint. ( notice this is possible because Period Q = Period - Q !!! Example f ' (A) = 1/ f ' (B) ) Now we construct a function that can be used as a kind of slog type , including if necc BRANCHES for the FUNCTIONAL inverse to make iterations of f. Call the slog type function T Then we have for a suitable C whenever f is not growing too fast : ( C depends on the growth of f and the derivate at the fixpoints ) T(x) = ... Rf^[-2] c^2 + Rf^[-1] C + R ( x ) + Rf(x)/C + R f^(x)/c^2 + ... Where Rf^[y] means R ( f^[y] (x) ) and R(x) = (x-a)(x-b). Now we split Up t(x) into t1(x) + t2(x) in the obvious way ( separate positive and negative iterates ) Such that we arrive at T(x) = t1(x) + t2(x) T(f(x)) = t1(x) C + t2(x) / C. So finally ( again for inverse use branches ) We get the slog type equation T( f(x)^[y] ) = t1(x) C^y + t2(x) C^{-y} As desired. Since the periods agree and we can use analytic continuation we have a solution that is analytic in the interval [A, +oo] , assuming that T is analytic in a nonzero radius. So we have a solution agreeing on 2 fixpoints !! Regards Tommy1729 « Next Oldest | Next Newest »

 Possibly Related Threads... Thread Author Replies Views Last Post Perhaps a new series for log^0.5(x) Gottfried 3 892 03/21/2020, 08:28 AM Last Post: Daniel My interpolation method  tommy1729 1 414 02/20/2020, 08:40 PM Last Post: tommy1729 Kneser method question tommy1729 9 1,784 02/11/2020, 01:26 AM Last Post: sheldonison Merged fixpoints of 2 iterates ? Asymptotic ?  tommy1729 1 1,019 09/10/2019, 11:28 AM Last Post: sheldonison Half-iterates and periodic stuff , my mod method  tommy1729 0 674 09/09/2019, 10:55 PM Last Post: tommy1729 Taylor series of i[x] Xorter 12 13,807 02/20/2018, 09:55 PM Last Post: Xorter b^b^x with base 0

Users browsing this thread: 1 Guest(s)