Solution 5 (b).

See text and/or instructor's solution manual.

Answer  .  

                    The point    that lie inside the contour  .

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

The integrand    is analytic everywhere except at the points    that lies inside the contour  .

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

                    .  
                    
Since the point    lies outside the contour  ,     is analytic inside and on  ,   

and the Cauchy-Goursat Theorem is used to evaluate the second integral:
                    
                    

Now use Corollary 6.1 to evaluate the integral:   .  

Therefore, we have  

                       

We are done.   

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

However, the integrand must first be expressed in the special form  , where  F(z) is analytic inside and on C.

In Section 6.5, we will be able revisit this exercise, using  ,  and the contour  ,  and the careful observation that  

                    ,   where   ,   and   .

Here    is analytic inside and on    and we are permitted to use the Cauchy Integral Formula  

                    .  

We are really done.   

        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 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