# Jordan's Lemma

Here we follow standard texts, such as Spiegel (1964)^{[1]} or Levinson and Redheffer (1970). ^{[2]}

In the computation of Fourier transform-like integrals as contour integrals, we often encounter the issue of the contribution of a semicircular contour that has infinite
radius.

## Statement of Jordan's lemma

Given a complex valued function such that as . This means
that as becomes large. We require also that
be analytic at infinity in some half-plane of
we have the result that the integral over a semicircular contour in that half-plane vanishes

The quantity is real valued, and finite.

### Proof of Jordan's lemma

Following the hypothesis of the lemma we consider the following contour integral

If we let and note by hypothesis that on , we have

where the last equality follows because the integrand is symmetric about the point .

The function on the interval allowing the further estimate (see Figure 2)

The last integral may be solved by elementary means to yield

as .

Though we have depicted the contour of integration in the upper half-plane of , the result is actually general. The value of the parameter takes on the appropriate sign as to make the exponential always decaying. So, a similar result exists for closure of the contour in the lower half-plane of , for Fourier-like integrals. Furthermore, for Laplace-like integrals we may consider closure in the right or left half-planes of the integration variable and obtain a similar result.