= Fourier series of a half-wave rectified sinusoid
{c}
For $V(t)=V_0\max(\sin\omega t,0)$, with period $2\pi/\omega$, <half-wave rectification> gives
$$
V(t)=\frac{V_0}{\pi}+\frac{V_0}{2}\sin\omega t
-\frac{2V_0}{\pi}\sum_{m=1}^\infty\frac{\cos(2m\omega t)}{4m^2-1}.
$$
The <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>.
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