= Step response of a damped oscillator
{title2=$H(t)K(t),\quad K(t)=\int_0^t g(s)ds$}
Integrating the smooth positive-time impulse kernel gives the zero-data response to the <Heaviside step function>. For nonzero $\omega$,
$$
K(t)=\frac{1-e^{-kt}[\cos\omega t+(k/\omega)\sin\omega t]}{k^2+\omega^2}.
$$
At zero frequency and nonzero $k$ the limit is $[1-e^{-kt}(1+kt)]/k^2$; when both parameters vanish it is $t^2/2$. The integral representation avoids ambiguous divisions in these limiting cases.
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