**Proof of The Real Version**

A simple proof is as follows.

For the first term, we note that *ε*⁄_{π(x2 + ε2)} is a nascent delta function, and therefore approaches a Dirac delta function in the limit. Therefore, the first term equals ∓*i*π *f*(0).

For the second term, we note that the factor *x*2⁄(*x*2 + *ε*2) approaches 1 for |*x*| ≫ *ε*, approaches 0 for |*x*| ≪ ε, and is exactly symmetric about 0. Therefore, in the limit, it turns the integral into a Cauchy principal value integral.

Read more about this topic: Sokhotski–Plemelj Theorem

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