Differentiably redundant functions. I am looking for a differentiably redundant function of order 6 from the following.
(a) $e^{-x} + e^{-x/ 2} \cos({\sqrt{3x} \over 2})$
(b) $e^{-x} + \cos(x)$
(c) $e^{x/2}\sin({\sqrt{3x} \over 2})$
I know that (b) has order 4, but I cannot solve for (a) and (c).
It would be a huge waste of time if I took the derivatives and calculate them, so there must be a simple way to solve this.
According to the book, it is related to $1/2 \pm i\sqrt{3} /2$, but why is that?

it is obvious that e^(-1) is of order 2
exp(-x/2)cos(sqrt(3)/2 x) is corresponding to -1/2+-sqrt(3)/2, which are 3rd roots of unity (see ODE textbooks for the general solution of homogeneous linear ODE of high orders), and they generate a cyclic group that contains all 3rd roots of unity.
For A, lcm(2,3)=6, so its order is 6.
For B, lcm(2,4)=4, so its order is 4.
For C, exp(x/2)sin(sqrt(3)/2 x) is corresponding to 1/2+-sqrt(3)/2, which are 6th roots of unity, are the generators of cyclic group that contains all of 6th roots of unity. So its order is 6.
