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f(x) = f(x+2pi) = 0 + a1 x - a2 x^2 + a3 x^3 + a4 x^4 - a5 x^5 + ... - tommy1729 - 06/15/2014
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/tetrationforum/showthread.php?tid=882
regards
tommy1729
RE: f(x) = f(x+2pi) = 0 + a1 x - a2 x^2 + a3 x^3 + a4 x^4 - a5 x^5 + ... - tommy1729 - 06/15/2014
I was thinking about elliptic functions as candidates ...
regards
tommy1729