= Ordered Gaussian response integral
{title2=$\operatorname{Re}\int_{\mathbb R}dt\int_{-\infty}^tds\,e^{-i\omega(t-s)}g(t)g(s)=\pi\tau^2e^{-\omega^2\tau^2}$}
For $g(t)=e^{-(t-t_0)^2/(2\tau^2)}$, the real integrand is symmetric under $t\leftrightarrow s$, so the ordered domain contributes half the full plane. The result is $\tfrac12|\int g(t)e^{-i\omega t}\,dt|^2=\pi\tau^2e^{-\omega^2\tau^2}$, using the <Fourier transform> of a <Gaussian function>. The imaginary part is not generally zero and must not be discarded before pairing conjugate harmonics in a real response.
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