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$$ f\left( x^{\prime },t+\varepsilon \right) = \int_{-\infty }^\infty dx\int_{-i\infty }^{i\infty } \frac{d\tilde{x}}{2\pi i} \left(1+\varepsilon \left[ \tilde{x}D_{1}\left( x,t\right) +\tilde{x}^{2}D_{2}\left( x,t\right) \right] \right) e^{\tilde{x}\left(x-x^{\prime }\right) }f\left( x,t\right) +O\left( \varepsilon ^{2}\right) \\ =\int_{-\infty }^\infty dx\int_{-i\infty }^{i\infty }\frac{d\tilde{x}}{2\pi i}\exp \left( \varepsilon \left[ -\tilde{x}\frac{\left( x^{\prime}-x\right) } {\varepsilon }+\tilde{x}D_{1}\left( x,t\right) +\tilde{x}^{2}D_{2}\left( x,t\right) \right] \right) f\left( x,t\right) +O\left(\varepsilon ^{2}\right).$$

Let's say we iterate $(t' - t)/\varepsilon$ times and perform the limit $\varepsilon \longrightarrow 0$.

How would we be able to calculate to the final form of $$L=\int dt\left[ \tilde{x}D_1 \left( x,t\right) +\tilde{x}^{2}D_2 \left( x,t\right) -\tilde{x}\frac{\partial x}{\partial t}\right].$$?


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