= Causal Green function of an overdamped oscillator
{title2=$G(t)=H(t)\frac{e^{-bt}-e^{-at}}{a-b},\quad a>b>0$}
For $L=(D_t+a)(D_t+b)$ with $a>b>0$, $H$ denotes the <Heaviside step function>. This <Green function> vanishes at negative time and satisfies $LG=\delta(t)$. Its positive-time part solves the homogeneous <ordinary differential equation>, is zero at the origin, and has right <derivative> one. Thus it is continuous with a unit derivative jump, which gives precisely the <Dirac delta function> in its second <distributional derivative>.
With the negative-exponential <Fourier transform> convention,
$$
\widehat G(k)=\frac1{(a+ik)(b+ik)}=\frac1{a-b}\left(\frac1{b+ik}-\frac1{a+ik}\right).
$$
Both <poles> lie in the upper half-plane. <Contour integration> closes above for positive time and below for negative time, giving the title expression. When the integral converges, <convolution> $y(t)=\int_{-\infty}^tG(t-s)f(s)ds$ supplies the response with no freely added homogeneous transient growing toward negative infinity. The continuous limit $a\to b$ is $G(t)=H(t)te^{-bt}$.
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