= Causal Green function of a damped oscillator
{title2=$G(t)=H(t)e^{-kt}\sin(\omega t)/\omega$}
= Damped-oscillator impulse response
{synonym}
For $L=D_t^2+2kD_t+k^2+\omega^2$, the causal <Green function> satisfies $LG=\delta(t)$. It is continuous at zero, with a unit jump in its first derivative. Its smooth positive-time kernel $g$ obeys $g(0)=0,g'(0)=1$. The zero-frequency limit is $H(t)te^{-kt}$. Shifting $G$ gives the zero-initial-data response to a delayed <Dirac delta function>.
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