Sokhotski–Plemelj Theorem - Proof of The Real Version

Proof of The Real Version

A simple proof is as follows.

lim_{varepsilonrightarrow 0^+} int_a^b frac{f(x)}{xpm i varepsilon},dx = mp i pi lim_{varepsilonrightarrow 0^+} int_a^b frac{varepsilon}{pi(x^2+varepsilon^2)}f(x),dx + lim_{varepsilonrightarrow 0^+} int_a^b frac{x^2}{x^2+varepsilon^2} , frac{f(x)}{x}, dx.

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 ∓iπ f(0).

For the second term, we note that the factor x2⁄(x2 + ε2) 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.

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