Exercise 11.  ,  where C is the line segment from  .  

Solution 11.

See text and/or instructor's solution manual.

Answer.  .  

Solution.  The function    is analytic on the entire complex plane, except points on the negative x-axis and at the origin,

and    is analytic in the right half-plane that contains the  line segment joining  .  

and    has    is an antiderivative.

Thus,  we can use Theorem 6.9 to obtain  

                              
                              

                    The path of integration is the line segment from  .  

                    Notice that    is not analytic on it's branch cut   .

We are 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