Product of square of distances from vertices of a polygon of radius a

Question

I want to find out the following product. $\prod_0^{n-1} (r^2 + a^2 -2ra\cos(2k\pi/n - \theta))$ I have been trying to use complex numbers but did not work out. The book I am reading the result is directly stated as $r^{2n} + a^{2n} -2r^na^n\cos(\theta)$. Please help.

Answer 1

Your product is the product $P$ of the squares of the distances from $z=re^{i\theta}$ to the points $ae^{2k\pi i/n}$ for $n=0,1,\ldots,n-1$. We have $$\eqalign{P &=\prod_{k=0}^{n-1}\bigl|z-ae^{2k\pi i/n}\bigr|^2\cr &=\biggl|\,\prod_{k=0}^{n-1}(z-ae^{2k\pi i/n})\biggr|^2\cr &=|z^n-a^n|^2\cr &=(r^ne^{ni\theta}-a^n)(r^ne^{-ni\theta}-a^n)\cr &=r^{2n}+a^{2n}-2r^na^n\cos n\theta\ .\cr}$$

Comment. From your question, I should think that you started by writing the modulus $|z-ae^{2k\pi i/n}|$ in terms of real numbers. Observe that it is easier if you do as much simplification as you can with complex numbers first, and switch to reals afterwards.