= Causal Green function of a damped oscillator
{title2=$G(t)=H(t)e^{-kt}\sin(\omega t)/\omega$}
= Damped-oscillator impulse response
{synonym}
For $L=D_t^2+2kD_t+k^2+\omega^2$, with $k>0$ and $\omega>0$, the causal <Green function> is $G(t)=H(t)e^{-kt}\sin(\omega t)/\omega$, where $H$ is the <Heaviside step function>. It 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>.
With the <Fourier transform> convention $\widehat f(\xi)=\int f(t)e^{-i\xi t}\,dt$, the transformed equation gives $\widehat G(\xi)=[k^2+\omega^2-\xi^2+2ik\xi]^{-1}$. Its <poles> $\xi=ik\pm\omega$ lie in the upper half of the <complex plane>. Inverting by <contour integration> therefore gives zero for negative time and an exponentially damped oscillation for positive time.
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