8.4  Improper Integrals Involving Trigonometric Functions

    Let P(x) and Q(x) be polynomials of degree m and n, respectively, where  .  We can show (but omit the proof) that if    for all real x, then  

                and      

are convergent improper integrals.  You may encounter integrals of this type in the study of Fourier transforms and Fourier integrals.  We now show how to evaluate them.

    Particularly important is our use of the identities

                and      

where    is a positive real number.  The crucial step in the proof of Theorem 8.4 wouldn't hold if we were to use    and    are used instead of  , as you will see when you get to Lemma 8.1.

Theorem 8.4 (Contour Integration for Improper Trig. Integrals).  Let P(x) and Q(x) be polynomials with real coefficients, of degree m and n, respectively, where    and    for all real x .  If    and  

(8-12)            ,    then

(8-13)            ,    and  
    
(8-14)            ,
  
where    are the poles of f(z) that lie in the upper half plane and    and    are the real and imaginary parts of  ,  respectively.  

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

Proof.

Proof of Theorem 8.4 is in the book.
Complex Analysis for Mathematics and Engineering

    The proof of Theorem 8.4 is similar to the proof of Theorem 8.3.  Before turning to the proof, we illustrate how to use Theorem 8.4.

Example 8.17.  Evaluate  .  

Solution.  The function f(z) in Equation (8-12) is  ,  which has a simple pole at the point    in the upper half-plane.  Calculating the residue yields

               

Using Equation (8-14) gives

              

Explore Solution 8.17.

Example 8.18.  Evaluate  .  

Solution.  The function f(z) in Equation (8-12) is  ,  which has simple poles at the points    and    in the upper half-plane.  We get the residues with the aid of L'Hôpital's rule:  

              

Similarly,

            

Using Equation (8-13), we get  

              

Explore Solution 8.18.

    We are almost ready to give the proof of Theorem 8.4, but first we need one preliminary result known as Jordan's lemma.  

Lemma 8.1, (Jordan's Lemma).  Suppose that P(z) and Q(z) are polynomials of degree m and n, respectively, where  .  If    is the upper semicircle ,  then

            .  

Proof.

Proof of Lemma 8.1 is in the book.
Complex Analysis for Mathematics and Engineering