# Find minimum maximum of sum of absolute values of sines, offset by equidistant phases

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.