 Fourier series of a half-wave rectified sinusoid

For $V(t)=V_0\max (\sin \omega t,0)$, with period $2\pi /\omega$, half-wave rectification givesThe Fourier coefficients are $a_0=2V_0/\pi$, $b_1=V_0/2$, $a_{2m}=-2V_0/[\pi (4m^2-1)]$, with all remaining coefficients zero. To derive the cosine coefficients, integrate $\sin s\cos ns$ over $0<s<\pi$ and use the product-to-sum identity. The sine coefficients follow from orthogonality. A half-period delay reverses only the fundamental sine term and leaves the constant and even-frequency cosine terms unchanged, by translation of a Fourier series.