display | more...

Jordan's Lemma is a small but important mathematical result that is useful in contour integration. In complex analysis we often wish to integrate functions around large semicircles in the complex plane, and Jordan's lemma provides useful information about how these integrals behave.

Statement of result

Let

J = integral(g(z)exp(ikz)dz, z=Rexp(iθ), θ=0...π)

where k>0, and |g(z)| tends to zero as |z| gets large. Then J tends to zero 0 as R tends to infinity.

Proof

First define

M(R) = max(|g(Rexp(iθ))|, θ=0...π)

Since |g(z)| tends to zero as |z| gets large, we deduce that M(R) tends to zero as |z| tends to zero. Hence, using the identity sin(θ) >= 2θ/π for 0 <= θ <= π/2, we have

|J| ≤ integral(|g(Rexp(iθ)|exp(-kRsin(θ))Rdθ, θ=0...π)
    ≤ R M(R) integral(exp(-kRsin(θ))dθ, θ=0...π)
    = 2R M(R) integral(exp(-kRsin(θ))dθ, θ=0...π/2)
    ≤ 2R M(R) integral(exp(-kR 2θ/π)dθ, θ=0...π/2)
    = R M(R) π/(kR) (1 - exp(-kR))
    ≤ M(R) π/k,

which tends to zero as R increases, as required.

Comments

At first sight the result seems almost trivial, since exp(ikz) exhibits exponential decay in the upper half plane. However, we must be careful, since as R increases, the length of the path increases; and the integrand does not decay exponentially near the ends of the path.

Another reason for concern is that if we view the integrand as a function on the Riemann Sphere, then it has an essential singularity at infinity, and many of the standard results of complex analysis may not hold.

Consequently we must proceed with caution. The are many examples of integrals similar to those in Jordan's Lemma which look as though they should tend to zero, but in fact do not.

Naturally, a corresponding result exists for the lower half plane.