8.5  Indented Contour Integrals

    If f(x) is continuous on the interval  , but discontinuous at b, then the improper integral of f(x) over is defined by  

            

provided that the limit exists.  Similarly, if f(x) is continuous on the interval  , but discontinuous at b, then the improper integral of f(x) over is defined by  

            

provided that the limit exists.  For example,  

            .

    If we let f(x) be continuous for all values of x in the interval ,  except at the value ,  where .  The Cauchy principal value of f(x) over is defined by
    
            ,  
    
provided that the limit exists.

Example 8.19.  Evaluate  .  

Solution.  Evaluating the integrals and computing limits gives  

                

Explore Solution 8.19.

    In this section we show how to use residues to evaluate the Cauchy principal value of the integral of f(x) over when the integrand f(x) has simple poles on the x axis.  We state our main results and then look at some examples before giving proofs.

Theorem 8.5.  Let where P(x) and Q(x) are polynomials with real coefficients of degree m and n, respectively, and  .  If Q(x) has simple zeros at the points on the x-axis, then  

(8-20)            ,  

where are the poles of f(z) that lie in the upper half-plane.  

Proof.

Proof of Theorem 8.5 is in the book.
Complex Analysis for Mathematics and Engineering

Figure 8.6  The poles of that lie on the x-axis and the poles that lie above the semicircles .

Theorem 8.6.  Let P(x) and Q(x) be polynomials, of degree m and n, respectively, where and let Q(x) have simple zeros at the points    on the x-axis.  If is a positive real number and if  , then we can compute the Cauchy Principal Value (P.V.) of the following integrals

(8-21)            ,  
    
        and  
    
(8-22)            ,    

where are the poles of f(z) that lie in the upper half plane.  

Proof.

Proof of Theorem 8.6 is in the book.
Complex Analysis for Mathematics and Engineering

Remark. The formulas in these theorems give the Cauchy principal value of the integral, which pays special attention to the manner in which any limits are taken.  They are similar to those in Section 8.3 and Section 8.4, except here we add one-half of the value of each residue at the points on the x-axis.

Example 8.20.  Evaluate    by using complex analysis.  

Solution.  The integrand  

              

has simple poles at the points    on the x axis and    in the upper half-plane.  Computing the residues we get

              
            
            

By Theorem 8.5,  

                

Explore Solution 8.20.

Example 8.21.  Evaluate    by using a computer algebra system.  

Solution.  Computer algebra systems such as Mathematica or MAPLE give the indefinite integral   

            .  

However, for real numbers, we should write the second term as    and use the equivalent formula:  

                

Figure 8.5  Graph of  .

    This antiderivative has the property that  ,  as shown in Figure 8.5.  We also compute  

              

              

The Cauchy principal limit at    as    is  

  

Therefore the Cauchy principal value of the improper integral is  

           

Explore Solution 8.21.

Example 8.22.  Show that  .    

Solution.  The integrand    has simple poles at the points    on the x axis and    in the upper half-plane.  Computing the residues we get  

              

            

By Theorem 8.6,

               

Explore Solution 8.22.

    The proofs of Theorems 8.5 and 8.6 depend on the following result.

Lemma 8.2.  Let f(z) have a simple pole at the point    on the x-axis.  If the contour is  ,  then

        .  

Proof.

Proof of Lemma 8.2 is in the book.
Complex Analysis for Mathematics and Engineering

Exercises for Section 8.5.  Indented Contour Integrals