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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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The diagram shown in the graphic gives a pictorial description of this method for evaluating integrals.

In this Demonstration, you are challenged to apply Jordan's lemma in 10 problems of increasing difficulty. Solutions are given.

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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.

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