 Fourier series of a noninteger-frequency cosine

Extend $\cos (\mu x)$ from $[-\pi ,\pi ]$ with period $2\pi$. It is an even function, so all sine Fourier coefficients vanish. The product-to-sum identity givesIndeed, integrate $\cos (\mu x)\cos (nx)$ on $[0,\pi ]$ and use $\sin (\pi (\mu \pm n))=(-1{)}^n\sin (\pi \mu )$. ThusThe coefficients are $O(n^{-2})$, so the series converges absolutely and uniformly. The Dirichlet-Jordan convergence theorem identifies it with the periodic function everywhere, including the endpoints because the two endpoint values agree. Substituting a noninteger rational frequency and a convenient $x$ yields explicit series sums. Integer frequencies require taking removable limits rather than substituting into a zero denominator.