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: enter image description here

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.


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