• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Taylor polynomial. System of equations for the coefficients. tommy1729 Ultimate Fellow     Posts: 1,491 Threads: 355 Joined: Feb 2009 05/01/2015, 09:43 PM Assuming a_n Goes to < (2/3)^n ; ( this gives us a sufficiently Large radius such that the equation is satisfied within the ROC.) Taylors theorem gives us f(x+1) = f(x) + f ' (x) + f " (x)/2 + ... Hence what the truncation of degree k solves locally is near ; f(x+1) + O(a_k x^k) = exp(f(x)) f(0)=1 By taking k Large and x small we get : f(0)=1 f(x+1)=exp(f(x)) + o(f(1)). ( take x < 1 to see this ) Notice lim o(f(1)) = lim a_k = 0. Hence we have in the limit k to oo assuming the ROC ; f(0)=1 f(x+1) = exp(f(x)) Qed So the attention Goes completely to the asymp of a_n. Hope that is clear. Q: Can we show existance and uniqueness for these equations FORMALLY ? Q2 ; i Will post in a new thread. Regards Tommy1729 « Next Oldest | Next Newest »

 Messages In This Thread Taylor polynomial. System of equations for the coefficients. - by marraco - 04/30/2015, 03:24 AM RE: Taylor polinomial. System of equations for the coefficients. - by tommy1729 - 05/01/2015, 08:37 AM RE: Taylor polinomial. System of equations for the coefficients. - by marraco - 05/01/2015, 09:42 AM RE: Taylor polinomial. System of equations for the coefficients. - by tommy1729 - 05/01/2015, 09:43 PM RE: Taylor polinomial. System of equations for the coefficients. - by marraco - 05/03/2015, 04:46 AM RE: Taylor polinomial. System of equations for the coefficients. - by marraco - 05/03/2015, 12:07 PM RE: Taylor polinomial. System of equations for the coefficients. - by Gottfried - 05/05/2015, 07:40 AM RE: Taylor polinomial. System of equations for the coefficients. - by marraco - 05/06/2015, 02:42 PM RE: Taylor polinomial. System of equations for the coefficients. - by Gottfried - 05/06/2015, 04:17 PM RE: Taylor polynomial. System of equations for the coefficients. - by marraco - 05/07/2015, 09:45 AM RE: Taylor polynomial. System of equations for the coefficients. - by marraco - 01/14/2016, 12:47 AM

 Possibly Related Threads... Thread Author Replies Views Last Post Arbitrary Order Transfer Equations JmsNxn 0 636 03/16/2021, 08:45 PM Last Post: JmsNxn New Quantum Algorithms (Carleman linearization) Finally Crack Nonlinear Equations Daniel 2 1,435 01/10/2021, 12:33 AM Last Post: marraco Moving between Abel's and Schroeder's Functional Equations Daniel 1 3,030 01/16/2020, 10:08 PM Last Post: sheldonison Taylor series of i[x] Xorter 12 23,071 02/20/2018, 09:55 PM Last Post: Xorter Taylor series of cheta Xorter 13 25,258 08/28/2016, 08:52 PM Last Post: sheldonison Totient equations tommy1729 0 3,326 05/08/2015, 11:20 PM Last Post: tommy1729 Bundle equations for bases > 2 tommy1729 0 3,389 04/18/2015, 12:24 PM Last Post: tommy1729 Grzegorczyk hierarchy vs Iterated differential equations? MphLee 0 3,554 01/03/2015, 11:02 PM Last Post: MphLee A system of functional equations for slog(x) ? tommy1729 3 8,129 07/28/2014, 09:16 PM Last Post: tommy1729 partial invariant equations ? tommy1729 0 3,244 03/16/2013, 12:32 AM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)