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Unexpected condition on convergence of integral


Integrate[1 / (1 + (x - y)^2), {y, -Infinity, Infinity}] yields If[Im[x] != 0, Pi, ...], but (I think) this integral should converge to Pi unconditionally (try it for some real values of x).

Using the option "Assumptions -> Im[x] == 0" in Integrate[] further yields the following warning: "Integral of [bla] does not converge on {-Infinity, Infinity}".

Can anyone shed some light on why Integrate can't determine that this integral converges for all x?


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