= Fourier series of a noninteger-frequency sine
{title2=$\sin(\lambda x)\sim\sum_{n\geq1}\frac{2(-1)^{n+1}n\sin(\pi\lambda)}{\pi(n^2-\lambda^2)}\sin(nx)$}
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 $\sum_{n\geq1}n^2/(4n^2-1)^2=\pi^2/64$. Integer frequencies instead already belong to the ordinary sine basis and should not be substituted into the removable coefficient singularities without taking limits.
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