Application of Jordan's Lemma
Jordan's lemma yields a simple way to calculate the integral along the real axis of functions f (z) = eiazg(z) holomorphic on the upper half-plane and continuous on the closed upper half-plane, except possibly at a finite number of non-real points z1, z2, ..., zn. Consider the closed contour C, which is the concatenation of the paths C1 and C2 shown in the picture. By definition,
Since on C2 the variable z is real, the second integral is real:
The left-hand side may be computed using the residue theorem to get, for all R larger than the maximum of |z1|, |z2|, ..., |zn|,
where Res(f, zk) denotes the residue of f at the singularity zk. Hence, if f satisfies condition (*), then taking the limit as R tends to infinity, the contour integral over C1 vanishes by Jordan's lemma and we get the value of the improper integral
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