Solution 9.

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  

                              .

This solution is complements of the authors.

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