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A calculus proposition about sum and derivative - tommy1729 - 08/19/2016
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 RE: A calculus proposition about sum and derivative - tommy1729 - 08/19/2016
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 |