= Residual amplitude of an oscillator after an integrable force
{title2=$R=\omega^{-1}|\int_0^\infty F(s)e^{-i\omega s}ds|$}
Let $y''+\omega^2y=F(t)$ with $\omega>0$, zero initial displacement and <velocity>, and real-valued forcing with $\int_0^\infty|F(t)|dt<\infty$. The causal <Green function> gives $y(t)=\omega^{-1}\int_0^tF(s)\sin[\omega(t-s)]ds$. Expanding the sine difference, the two coefficients tend to finite limits by <absolute convergence>. Thus $y(t)$ approaches a sinusoid whose amplitude is the displayed <modulus> of the one-sided <Fourier transform>, divided by $\omega$. The displacement error is at most $\omega^{-1}\int_t^\infty|F(s)|ds$, which tends to zero. This describes the surviving free oscillation after a transient force; an arbitrary nonzero initial oscillation must be added before taking the final amplitude.
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