Solution 12 (b).

See text and/or instructor's solution manual.

Answer.  .

                    The points    that lie inside the contour  .  

Solution.  For this part of the exercise there are two points that lie inside the circle .

Both Cauchy's Integral formula and Cauchy's Integral formula for derivatives apply when only one point lies inside the circle .

Therefore, we need to split the integrand into two parts  ,  and the integral can be written as  

                    

Then we have two integrals to compute:      and   

Solution part (a).  Show that  .

                    The point    that lies inside the contour  .  

Here the integrand   is not defined at the point    which lies interior to the circle  ,

and the integral    has the form    so we can use the Cauchy Integral Formula (see Section 6.5).

Here we have    and calculation reveals that  .  

Applying the Cauchy's integral formula    we write   

                              .  

Then multiplication by    establishes the desired result  

                              .

Solution part (b).  Show that  .

                    The point    that lies inside the contour  .  

Here the integrand   is not defined at the point    which lies interior to the circle  ,

and the integral    has the form    so we can use the Cauchy Integral Formula (see Section 6.5).

Here we have    and calculation reveals that  .  

Applying the Cauchy's integral formula    we write   

                              .  

Then multiplication by    establishes the desired result  

                              .

Now we combine the two results from Solution part (a) and solution part (b).  

Therefore  .

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

                       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:

                      

          and

                      

We are really really done.   

Aside.  We can let Mathematica double check our work.

Remark.  Since      and     

the calculation with Cauchy's integral formula 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