8.6  Integrands with Branch Points

    We now show how to evaluate certain improper real integrals involving the integrand  .  Since the complex function    is multivalued, so we must first specify the branch to be used.

    Let be a real number with .  In this section we use the branch of corresponding to the branch of the logarithm (see Equation (5-20)) as follows:

            ,  

where    and  .  Note that this is not the traditional principal branch of and that, as defined, the function is analytic in the domain .

Theorem 8.7.  Let P(x) and Q(x) be polynomials of degree m and n, respectively, where  .  If  ,  and Q(x) has a zero of order at most 1 at the origin, and , then  

            ,  

where    are the non-zero poles of  .  

Figure 8.7  The contour C that encloses the nonzero poles    of  .

Proof.

Proof of Theorem 8.7 is in the book.
Complex Analysis for Mathematics and Engineering

Example 8.23.  Evaluate  ,  where  .  

Solution.  The function    has a nonzero pole at the point , and the denominator has a zero of order at most 1 (in fact, exactly 1) at the origin.  Calculating the residue we get  

            

Using Theorem 8.7, we have  

               

Explore Solution 8.23.

    We can apply the preceding ideas to other multivalued functions.  

Example 8.24.  Evaluate  ,  where  .  

Solution.  We use the function  .  Recall that  

            ,  


where    and  .  The path C of integration will consist of the segments of the x axis together with the upper semicircles    and  ,  for  ,  as shown in Figure 8.8.

Figure 8.8  The contour C for the integrand  .

    We chose the branch    because it is analytic on C and its interior, hence so is the function f(z) which has a simple pole in the upper half-plane at the point .

            

This choice enables us to apply the residue theorem properly (see the hypotheses of Theorem 8.1), and we get  

              

Keeping in mind the branch of logarithm that we're using, we then have  

(8-33)               

If  ,  then by the ML inequality (Theorem 6.3)

            

and L'Hôpital's rule yields  .  A similar computation shows that  .  
We use these results when we take limits Equations (8-33) to get  

               

Equating the real parts in this equation gives  

            

Explore Solution 8.24.

Exercises for Section 8.6.  Integrands with Branch Points