example of using residue theorem
We take an example of applying the Cauchy residue theorem in evaluating usual real improper integrals.
We shall the integral
Since the integrand is an even (http://planetmath.org/EvenoddFunction) and an odd function, we may write
using also Euler’s formula. Let’s consider the contour integral
where is the perimeter of the semicircle consisting of the line segment from to and the semi-circular arc connecting these points in the upper half-plane (). The integrand is analytic on and inside of except in the point which is a simple pole. Because we have (cf. the coefficients of Laurent series)
the residue theorem (http://planetmath.org/CauchyResidueTheorem) yields
This does not depend on the radius of the circle.
We split the integral in two portions: one along the diameter and the other along the circular arc . So we obtain
When , the former portion tends to the limit and the latter — as we at once shall see — to the limit 0. Hence we get the result
As for the latter part of , we denote (); then on the arc , where and , we have
Since the right hand member tends to 0 as , then also the left hand member.
|Title||example of using residue theorem|
|Date of creation||2013-03-22 15:19:30|
|Last modified on||2013-03-22 15:19:30|
|Last modified by||pahio (2872)|