Finite-endpoint branch-cut jump integral ($F(-x-i0)-F(-x+i0)$)

A pair of oppositely directed contours on a branch cut produces the difference of the two boundary values integrated over their actual endpoint interval. Replacing finite endpoints by zero and infinity requires a limiting argument and convergence of the combined improper integral. Individual side integrals may diverge even when their difference converges, so cancellation must precede separation of endpoint limits.