8.3  Improper Integrals of Rational Functions

    An important application of the theory of residues is the evaluation of certain types of improper integrals.  Let f(x) be a continuous function of the real variable x on the interval  .  Recall from calculus that the improper integral of f(x) over  [)  is defined by

            ,  
    
provided that the limit exists.  If f(x)  is defined for all real x, then the integral of f(x) over is defined by

(8-7)            ,  

provided both limits exist.  If the integral in Equation (8-7) exists, we can obtain its value by taking a single limit:

(8-8)            .  

For some functions the limit on the right side of Equation (8-8) exists, but the limit on the right side of Equation (8-7) doesn't exist.

Example 8.13.  ,  

but Equation (8-7) tells us that the improper integral of    over   doesn't exist, because the calculation  

                is undefined .  

Therefore we can use Equation (8-8) to extend the notion of the value of an improper integral, as Definition 8.2 indicates.  

Explore Solution 8.13.

Extra Example 1.  ,  

and Equation (8-7) tells us that the improper integral of    over   also diverges to , because  

            .  

Explore Extra Solution 1.

Definition 8.2 (Cauchy Principal Value - P.V.).  Let f(x) be a continuous real valued function for all x.  The Cauchy principal value (P.V.) of the integral    is defined by

            ,  

provided the limit exists.

    Example 8.13 shows that  .  

Example 8.14.  Find the Cauchy principal value of  .

Solution.  
           

Explore Solution 8.14.

    If  ,  where P(x) and Q(x) are polynomials, then f(x) is called a rational function.  Techniques in calculus were developed to integrate rational functions.  We now show how the Residue Theorem can be used to obtain the Cauchy principal value of the integral of f(x) over  .  

Theorem 8.3 (Contour Integration for Rational Functions).  Let    where P(x) and Q(x) are polynomials, of degree m and n , respectively.  If for all real x and  ,  then the Cauchy Principal Value (P.V.) of the integral is

            .  

where    are the poles of    that lie in the upper half plane.   The situation is illustrated in Figure 8.4.  

Figure 8.4  The poles    of    that lie in the upper half-plane.

Proof.

Proof of Theorem 8.3 is in the book.
Complex Analysis for Mathematics and Engineering

Example 8.15.  Evaluate  .  

Solution.  We write the integrand as  .  We see that f(z) has simple poles at the points and ,  and that the points    and  ,  are the only singularities of f(z) in the upper half-plane.  Computing the residues, we obtain  

              

               

Using Theorem 8.3, we conclude that  

              

Explore Solution 8.15.

Example 8.16.  Evaluate  .  

Solution.  The integrand    has a poles of order 3 at the points , and    is the only singularity of f(z) in the upper half-plane.  Computing the residue at   , we get  

               

Therefore

            .  

Explore Solution 8.16.