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.
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.
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