Solution 13 (a).

See text and/or instructor's solution manual.

Answer  .  

Solution.  Factor the denominator of the integrand    to see that  .  

The integrand    is analytic everywhere except at the points    which lie inside the contour  .

Now use partial fractions and express the integrand as  ,   and split up the integral into two parts

                     
                   
Notice that the contour    is negatively oriented and the contour    is positively oriented.

Use the Cauchy-Goursat Theorem to determine that      and   .

Now use Corollary 6.1 to evaluate the last two integrals:     and   
                    
Therefore, we have  

                       

                         Figure 6.28  The points    and    which lie inside the contour  .

We are done.   

Aside.   In Section 6.5 we will learn to evaluate contour integrals with the Cauchy Integral Formula.  

        The more advanced methods are easier to use but require special attention to details.  

Techniques for evaluating integrals using contours is an important area of complex analysis,

and we will explore this in detail starting with the Residue Calculus in Section 8.1.

        Once the Residue Calculus has been introduced in Section 8.1, computing this integral will be a trivial:

                     .

We are really done.   

Aside.  We can let Mathematica double check our work.

This solution is complements of the authors.

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