Jordan's lemma

Consider a complex-valued, continuous function f, defined on a semicircular contour of positive radius R lying in the upper half-plane, centered at the origin.

If the function f is of the form with a positive parameter a, then Jordan's lemma states the following upper bound for the contour integral: with equality when g vanishes everywhere, in which case both sides are identically zero.

An analogous statement for a semicircular contour in the lower half-plane holds when a < 0.

Jordan's lemma yields a simple way to calculate the integral along the real axis of functions f(z) = ei a z g(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.

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 The function satisfies the condition of Jordan's lemma with a = 1 for all R > 0 with R ≠ 1.

Since the only singularity of f in the upper half plane is at z = i, the above application yields Since z = i is a simple pole of f and 1 + z2 = (z + i)(z − i), we obtain so that This result exemplifies the way some integrals difficult to compute with classical methods are easily evaluated with the help of complex analysis.