# 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.

## References

^{[3]}

^{[4]}

^{[5]}

- ↑ Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).
- ↑ Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.
- ↑ Whaley, J., 2017, Oil in the Heart of South America, https://www.geoexpro.com/articles/2017/10/oil-in-the-heart-of-south-america], accessed November 15, 2021.
- ↑ Wiens, F., 1995, Phanerozoic Tectonics and Sedimentation of The Chaco Basin, Paraguay. Its Hydrocarbon Potential: Geoconsultores, 2-27, accessed November 15, 2021; https://www.researchgate.net/publication/281348744_Phanerozoic_tectonics_and_sedimentation_in_the_Chaco_Basin_of_Paraguay_with_comments_on_hydrocarbon_potential
- ↑ Alfredo, Carlos, and Clebsch Kuhn. “The Geological Evolution of the Paraguayan Chaco.” TTU DSpace Home. Texas Tech University, August 1, 1991. https://ttu-ir.tdl.org/handle/2346/9214?show=full.