# Jordan's Lemma Applied to the Evaluation of Some Infinite Integrals

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Jordan's lemma can be stated as follows: let be an analytic function in the upper half of the complex plane such that on any semicircle of radius in the upper half-plane, centered at the origin. Then, for , the contour integral as [1, 2]. This can be directly applied to the evaluation of infinite integrals of the form in terms of the residues of at the points in the upper half-plane. Specifically, . If is a pole of order , the residue is given by . The method is also applicable for , with the simpler integrals , provided that fulfills the requisite limiting behavior.

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Contributed by: S. M. Blinder (April 2018)

Open content licensed under CC BY-NC-SA

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When a singularity occurs on the real axis, such as with , mathematical physicists casually consider that *half* the pole is contained within the Cauchy region and that it contributes half the residue of that pole. Some pure mathematicians are very agitated by this shortcut, even though a more rigorous treatment involving an indented contour avoiding the singularity leads to the same result. The integral along the real axis actually represents the *Cauchy principal value*.

References

[1] E. T. Whittaker and G. N. Watson, *A Course of Modern Analysis*, 4th ed., Cambridge, UK: Cambridge University Press, 1958 pp. 115 ff.

[2] M. C. Jordan, *Cours d’Analyse de l’Ecole Polytechnique, Tome Deuxieme, Calcul Intégral*, Paris: Gauthier-Villars, 1894 pp. 285–286.