Exercise 15.  ,   where   .  

Solution 15.

See text and/or instructor's solution manual.

Answer.   .  

Solution.  The complex integrand is  .  

Factor the denominator and get

                    .  

It follows that    has poles of order  3  at  ,  

and the pole at    lies in the upper half plane.

Using Theorem 8.1 (Cauchy's Residue Theorem), and Theorem 8.3 (Contour Integration for Rational Functions), the value of the integral is computed   

                      

Here the denominator of  f(z) has a factor of the form  ,  and  .  

In this exercise, the limit can be calculated as follows:

                       

We are done.   

Aside.  We can let Mathematica double check our work.

Maple can check our work too!

     > residue( z^2/(z^2+a^2)^3, z=I*a );

               

     > 2*Pi*I*residue(z^2/(z^2+a^2)^3,z=I*a);

               

We are really done.   

Aside.  Both and are capable of finding the definite integral.

     > int( x^2/(x^2+a^2)^3, x=-infinity..infinity );

               

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell