For every integer $N>0$ given function $f_N(x) = \sum_0^{N-1} |\sin(x+\frac{2i\pi}{N})|$
Is there some $O(1)$ analytic solution (without using $\sum$ operation), to find its minimum $\min(f_N)=?$ and maximum $\max(f_N)=?$ values?
I could provide an example. For $N=3$ finite series function would expand to:
$$f_3(x) = |\sin(x)| + \left|\sin(x+\frac{\pi}{3})\right| + \left|\sin(x+\frac{2\pi}{3})\right|,$$
And its graph would be as follows:
It's now obvious that:
$\min(f_3) = f_3(0) = \sqrt{3}$
$\max(f_3) = f_3\left(\frac{\pi}{6}\right) = 2$
Now, for any $N$, I want to know analytic form of those $\min(f_N)$ and $\max(f_N)$. Goal is to obtain their analytic continuation for any real numbers (if it exists). Indeed, this could only be done without using summation $\sum$ operation. I believe that some simple formulas (just dependent on $N$) could exist.