Exercise 17.  Let    be the principal branch of the square root function.  

17 (b).  Evaluate  ,  where C is the right half of the circle    joining  .  

Solution 17 (b).

See text and/or instructor's solution manual.

Answer.  .  

Solution.  The function    is analytic everywhere except points   ,  

along the negative x-axis, where the principal branch of the square root function is discontinuous,

and    has    as an antiderivative.  

Letting D be the simply connected domain consisting of the entire complex plane except for the real numbers   ,  

along the negative x-axis, we see that    and its listed antiderivative are analytic in D.  

Since the the right half of the circle    joining    is contained in D, Theorem 6.9 gives

                      

                    The path of integration is the right half of the circle    joining  .  

                    Notice that    is not analytic on the ray   .

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