Solution 18.

See text and/or instructor's solution manual.

Answer.  

                        

When   we get the following limits:

                       

Observe that this is the Cauchy's Integral formula for derivatives      when   .  

                    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    where    so we can use the Cauchy Integral Formula (see Section 6.5).

Here we have  .  

Applying the Cauchy's integral formula  .  

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    where    so we can use the Cauchy Integral Formula (see Section 6.5).

Here we have  .  

Applying the Cauchy's integral formula  .  

Then multiplication by    establishes the desired result  

                              .

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

Therefore

                        

This solution is complements of the authors.

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