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The first equation, $$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$ is actually a shorthand notation for its correct full form, which is $$\underset{\epsilon\rightarrow0^+}{lim}\int_{-\infty}^\infty\frac{f(x)}{x-x_0+i\epsilon}\,dx=P\int_{-\infty}^\infty\frac{f(x)}{x-x_0}\,dx-i\pi f(x_0)$$ and is valid for functions which are analytic in the ...


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Going from equation (0) to (1) is basically by writing the adjoint of the linear operator in the Fokker Planck equation (with a standard $L^2$ inner product). So the first equation is basically the following statements- $$ \dfrac{\partial f(x,t)}{\partial t}=\mathcal{L}_xf\text{ where }\mathcal{L}_x\equiv-\dfrac{\partial}{\partial ...



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