example of using residue theorem

We take an example of applying the Cauchy residue theorem in evaluating usual real improper integrals.

We shall the integralDlmfPlanetmath


where k is any real number.  One may prove that the integrand has no antiderivative among the elementary functionsMathworldPlanetmath if  k0.

Since the integrand is an even (http://planetmath.org/EvenoddFunction) and  xsinkx1+x2  an odd functionMathworldPlanetmath, we may write


using also Euler’s formula.  Let’s consider the contour integral


where γ is the perimeter of the semicircle consisting of the line segment from (-R, 0) to (R, 0) and the semi-circular arc c connecting these points in the upper half-plane (R>1).  The integrand is analytic on and inside of γ except in the point  z=i  which is a simple poleMathworldPlanetmathPlanetmath.  Because we have (cf. the coefficients of Laurent series)


the residue theorem (http://planetmath.org/CauchyResidueTheorem) yields

J= 2πie-k2i=πe-k.

This does not depend on the radius R of the circle.

We split the integral J in two portions:  one along the diameter and the other along the circular arc c.  So we obtain


When  R,  the former portion tends to the limit 2I and the latter — as we at once shall see — to the limit 0.  Hence we get the result


As for the latter part of J, we denote  z:=x+iy (x,y);  then on the arc c, where  |z|=R  and  y0, we have


Using this estimation of the integrand we get, according the integral estimating theorem, the inequalityMathworldPlanetmath


Since the right hand member tends to 0 as  R, then also the left hand member.

Title example of using residue theorem
Canonical name ExampleOfUsingResidueTheorem
Date of creation 2013-03-22 15:19:30
Last modified on 2013-03-22 15:19:30
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 19
Author pahio (2872)
Entry type Example
Classification msc 30E20
Related topic ImproperIntegral
Related topic ResidueDlmfPlanetmath
Related topic IntegralsOfEvenAndOddFunctions
Related topic UsingResidueTheoremNearBranchPoint