Proof: When , we have the previously proved case. For arbitrary and real number , let denote the closed rectangular contour , depicted in Fig.D.1.
Clearly, is analytic inside the region bounded by . By Cauchy's theorem , the line integral of along is zero, i.e.,
where and are real variables. In the limit as , the first piece approaches , as previously proved. Pieces and contribute zero in the limit, since as . Since the total contour integral is zero by Cauchy's theorem, we conclude that piece 3 is the negative of piece 1, i.e., in the limit as ,