4.2.3 Example: \(N = 5\), \(j = 2\)
For \(N = 5\) and \(j = 2\):
\begin{equation} S_2(\theta ) = \sum _{k=0}^4 \cos ^2\left(\theta + \frac{2\pi k}{5}\right) = \frac{5}{2} + \frac{1}{2}\sum _{k=0}^4 \cos \left(2\theta + \frac{4\pi k}{5}\right) = \frac{5}{2}. \end{equation}
7
The trigonometric sum vanishes since \(m = 2 {\lt} N = 5\).