Ordered Gaussian response integral ($\mathrm{Re}{\int }_{\mathbb{R}}dt{\int }_{-\infty }^{t}ds{e}^{-i\omega (t-s)}g(t)g(s)=\pi {\tau }^2{e}^{-{\omega }^2{\tau }^2}$)

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 $\frac{1}{2}|\int g(t)e^{-i\omega t}dt{|}^2=\pi {\tau }^2{e}^{-{\omega }^2{\tau }^2}$, 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.