= Causal response of a first-order relaxation equation
{title2=$G(t)=\alpha^{-1}e^{-t/\alpha}H(t)$}
For $\alpha>0$, the retarded <impulse response> of $\alpha y'+y=f$ is the displayed function, with $H$ the <Heaviside step function>. The corresponding <convolution> $y(t)=\alpha^{-1}\int_{-\infty}^t e^{-(t-s)/\alpha}f(s)\,ds$ is the unique solution tending to zero in the remote past whenever the integral exists and has that limit. For example, bounded forcing that tends to zero as $t\to-\infty$ satisfies these conditions. A general forcing need not: $f\equiv1$ admits no such solution. A unit-duration pulse with $\alpha=1$ gives $y=0$ before zero, $y=1-e^{-t}$ between zero and one, and $y=(1-e^{-1})e^{-(t-1)}$ thereafter.
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