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Q: Is this relation between divergent intergals justifiable?

AnixxGraf's book on hyperfunction theory says (page $36$) that $$\frac1{(x-i0)^n}=\frac{(-1)^{n-1}\pi i}{(n-1)!}\delta^{(n-1)}(x)+\operatorname{fp}\frac1{x^n},$$ while the table of Fourier transforms says that the Fourier transform of $x^n$ is $2\pi i^n\delta^{(n)}(x)$ (here the Fourier transform is...

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