information:filtering
Sokhotski–Plemelj theorem

The Sokhotski–Plemelj theorem, important in quantum mechanics, relates the delta function to the distribution p.v. 1 / x, the Cauchy principal value of the function 1 / x, defined by

⟨ p. v. 1 x, φ ⟩ = lim ε → 0 + ∫ | x | > ε φ (x) x d x. {\displaystyle \left\langle \operatorname {p.v.} {\frac {1}{x}},\varphi \right\rangle =\lim _{\varepsilon \to 0^{+}}\int _{|x|>\varepsilon }{\frac {\varphi (x)}{x}}\,dx.}

Sokhotsky's formula states that

lim ε → 0 + 1 x ± i ε = p. v. 1 x ∓ i π δ (x), {\displaystyle \lim _{\varepsilon \to 0^{+}}{\frac {1}{x\pm i\varepsilon }}=\operatorname {p.v.} {\frac {1}{x}}\mp i\pi \delta (x),}

Here the limit is understood in the distribution sense, that for all compactly supported smooth functions f,

∫ − ∞ ∞ lim ε → 0 + f (x) x ± i ε d x = ∓ i π f (0) + lim ε → 0 + ∫ | x | > ε f (x) x d x. {\displaystyle \int _{-\infty }^{\infty }\lim _{\varepsilon \to 0^{+}}{\frac {f(x)}{x\pm i\varepsilon }}\,dx=\mp i\pi f(0)+\lim _{\varepsilon \to 0^{+}}\int _{|x|>\varepsilon }{\frac {f(x)}{x}}\,dx.}