f(x) = f(x+2pi) = 0 + a1 x - a2 x^2 + a3 x^3 + a4 x^4 - a5 x^5 + ... tommy1729 Ultimate Fellow     Posts: 1,493 Threads: 356 Joined: Feb 2009 06/15/2014, 03:30 PM (This post was last modified: 06/15/2014, 03:32 PM by tommy1729.) Im looking for real entire functions f(x) such that : f(x) = f(x+2pi) = 0 + a1 x - a2 x^2 + a3 x^3 + a4 x^4 - a5 x^5 + ... where an > 0 and the pattern continues {+,+,-}. This comes from the idea of understanding the signs of the nth derivatives of certain functions , being tetration related or more classical functions. For instance sin and cos have the pattern {+,-} ( omitting 0's ). SO it seems natural to ask about the pattern {+,+,-}. Consider the many theta functions and fourier series that seems natural. I have not seen this question before. Maybe this is easy and there might even be an elementary f(x). Maybe calculus 101 or trig 101 etc. But here is how I tried to consider it. Btw this also comes from my lost notebook. f(x) = A(x) + B(x) A(x) = 0 + A1 sin(x) + A2 sin(2x) + A3 sin(3x) + ... B(x) = 0 + B1 cos(x) + B2 cos(2x) + B3 cos(3x) + ... A(x) = 0 + (A1 + A2 2^1 + A3 3^1 + ...) x^1/1! - (A1 + A2 2^3 + A3 3^3 + ...) x^3/3! + (A1 + A2 2^5 + A3 3^5 + ...) x^5/5! - ... (A1 + A2 2^(2k+1) + A3 3^(2k+1) + ...) x^(2k+1)/(2k+1)! B(x) = 0 + (B1 + B2 + B3 + ...) x^0/1! - (B1 + B2 2^2 + B3 3^2 + ...) x^2/2! + (B1 + B2 2^4 + B3 3^4 + ...) x^4/4! - ... (B1 + B2 2^(2k) + B3 3^(2k) + ...) x^(2k)/(2k)! Now I have an infinite system of equations/inequalities that reminds me of dirichlet series. Matrix methods ?? I noticed {+-+-+-}*{+--+--}={++---+} {+-+-+-}*{+---++}={++-++-} I thought it might help. Is it possible for such an f(x) to exist ? Or do we need that 50 % have sign + and 50% have sign - ? I vaguely remember someone saying related stuff about truncated Taylor series leading to some limitations. A related question is how fast the derivatives of a 2pi periodic function can grow ? Related : c_1 + c_2 2^k + c_3 3^k + ... c_n n^k ~ exp(exp(k)) ? If we allow c_n to be 0 often it seems we might get an approximation from the Taylor series of exp(x). Otherwise I dont know. I was inspired to share this idea because of the thread : http://math.eretrandre.org/tetrationforu...hp?tid=882 regards tommy1729 tommy1729 Ultimate Fellow     Posts: 1,493 Threads: 356 Joined: Feb 2009 06/15/2014, 03:35 PM I was thinking about elliptic functions as candidates ... regards tommy1729 « Next Oldest | Next Newest »