Here we follow standard texts, such as Spiegel (1964) or Levinson and Redheffer (1970). 
We consider integrals of the form , where is a closed simple contour as shown in Figure 1.
The function has a finite number of poles located inside the contour. As the function in the integrand is assumed to decay at least as fast as , where for points on .
"Figure 1: the contour C for a Fourier-like integration (upward closure only)"
By the Residue Theorem the value of this integral may be written as times the sum of
the residues of the integrand
Typically, the application of such a contour integral is to find the value of the integral along the real axis, which is the integral along .
The next step is to evaluate the integral over the contour and show that this vanishes. We consider the following estimate
By Cauchy's theorem we may assume that is a semi-circle, and may consider the polar form of ,
, and we may also consider in the exponential. Substituting
these into the and applying the fact that as the function yielding
where is the radius of the semi-circle For the integral to exist, the integrand must decay sufficiently rapidly. Thus, the Here, the Maximum Modulus theorem has been applied to generate the estimate of
We may further proceed with the estimate by recognizing that integral is twice the same integral, with upper integration
limit of and the extra factor of 2 is absorbed into the yielding
Here we have used the fact that on the interval we have .
The case of closure in the lower half plane of follows the same way, with the exception that the
Thus, the integral over is given by the expression
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- ↑ 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.