Jordan's Lemma - Application of Jordan's Lemma

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, ..., z_n. 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|, ..., |z_n|,

where Res(f, z_k) denotes the residue of f at the singularity z_k. 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