A calculus proposition about sum and derivative tommy1729 Ultimate Fellow Posts: 1,906 Threads: 409 Joined: Feb 2009 08/19/2016, 12:19 PM Hi Let f(x) , g1(x) , g2(x) , ... be analytic on [-1,1]. If for almost Every x e [-1,1] we have : Property A : If f(x) = g(x) = g1(x) + g2(x) + ... [ property I ] And If f ' (x) = g1 ' (x) + g2 ' (x) + ... [property II] Then f '' (x) = g1 '' (x) + g2 '' (x) + ... And by induction the n th derivative satisfies ( n is a positive integer ) f^(n) (x) = g1^(n) (x) + g2^(n) (x) + ... **** notice property I does not always imply property II ; example Foerier series for x. **** How to prove or disprove this ? What are Nice examples ? How about variations ? ( such as replacing first and second derivative with second and third ). Does this motivite the desire to work with a new type of series expansions ? Regards Tommy1729 tommy1729 Ultimate Fellow Posts: 1,906 Threads: 409 Joined: Feb 2009 08/19/2016, 12:24 PM A variant is including property III also where this is the analogue for 3 th derivative and then making the same conclusion. This resembles the squeezing theorem imho. Also the q-derivate is of intrest. Regards Tommy1729 « Next Oldest | Next Newest »

 Possibly Related Threads… Thread Author Replies Views Last Post Bridging fractional iteration and fractional calculus Daniel 8 377 04/02/2023, 02:16 AM Last Post: JmsNxn CONCLUSION 1 : conditions for nice tetration : second derivative !! tommy1729 6 333 02/23/2023, 12:56 PM Last Post: tommy1729 Derivative of the Tetration Logarithm Catullus 1 631 07/03/2022, 07:23 AM Last Post: JmsNxn On the first derivative of the n-th tetration of f(x) Luknik 5 2,139 10/28/2021, 11:53 AM Last Post: tommy1729 Semi-exp and the geometric derivative. A criterion. tommy1729 0 3,615 09/19/2017, 09:45 PM Last Post: tommy1729 How to find the first negative derivative ? tommy1729 0 3,629 02/13/2017, 01:30 PM Last Post: tommy1729 Derivative of exp^[1/2] at the fixed point? sheldonison 10 24,349 01/01/2016, 03:58 PM Last Post: sheldonison Derivative of E tetra x Forehead 7 17,634 12/25/2015, 03:59 AM Last Post: andydude Fractional calculus and tetration JmsNxn 5 14,778 11/20/2014, 11:16 PM Last Post: JmsNxn A derivative conjecture matrix 3 9,361 10/25/2013, 11:33 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)