Solution 3.

See text and/or instructor's solution manual.

Answer.  .  

                    The point    that lies inside the contour  .  

Solution.  The integrand   is not defined at the point    which lies interior to the circle  ,

and the integral    has the form    where   ,  

so we can use Cauchy's Integral formula for derivatives (see Section 6.5).

Here we have      and      and calculation reveals that   .  

Applying Cauchy's Integral formula for derivatives      with      we write   

                              .  

Then multiplication by    establishes the desired result  

                              .

We are done.   

Aside.  Be patient, it is our goal to develop an elegant and efficient method to compute this type of contour integral.  

The method is known as the "Residue Calculus," and is introduced in Section 8.1.  

Looking Forward.  Cauchy's Residue Theorem (see Theorem 8.1) will use the notion of   given in Definition 8.1.

For this problems we have chosen the details so that the integrand    is analytic inside C and on C,  

except at the point    that lie inside C.  Then Cauchy's Residue Theorem states

                              .  

Using the Residue Calculus.  Let us announce that in Section 8.1, we can compute    with  ,  and

                                  and    .
                              
Then the "Residue Calculus" gives a quick way to find the answer:

                              .  
            
After the concept of   is developed in Section 8.1, you will enjoy calculating integrals the "easy way."

We are really done.   

Aside.  We can let Mathematica double check our work.

Preview of calculating a residue.

In Section 8.1 we will show that if the denominator of F(z) has a factor of the form  ,  then  .  

In this exercise, the the limit is calculated as follows:

We are really really done.   

Aside.  We can let Mathematica double check our work.

Remark.  Since      and     

the calculation with Cauchy's Integral formula for derivatives is the same as the residue calculation using a limit.

This solution is complements of the authors.

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