Theorem: Lorentzian sum-rule integral = pi * gamma / 2 (residue theorem, contour in upper half-plane)

Theorem (Breit-Wigner integral canonical): for a Lorentzian line-shape (gamma/2)^2 / ((E - E_R)^2 + (gamma/2)^2) with positive width gamma, the integral over all real E evaluates exactly to pi * gamma / 2, independent of resonance centre…