Fourier series of a noninteger-frequency sine ($\sin (\lambda x)\sim {\sum }_{n\ge 1}\frac{2(-1{)}^{n+1}n\sin (\pi \lambda )}{\pi (n^2-{\lambda }^2)}\sin (nx)$)

For real noninteger $\lambda$, the periodic extension from $[-\pi ,\pi ]$ has the displayed Fourier series, obtained by the product-to-sum identity in the sine coefficient integral. It equals the original function in the open interval and zero at the periodic endpoint jumps. At $\lambda =1/2$, Parseval's identity gives ${\sum }_{n\ge 1}{n}^2/(4n^2-1{)}^2={\pi }^2/64$. Integer frequencies instead already belong to the ordinary sine basis and should not be substituted into the removable coefficient singularities without taking limits.