Exercise 11.  Construct branches of    that are analytic at all points in the plane except at points on the following rays.  

11 (a).  .  

Solution 11 (a).

See text and/or instructor's solution manual.

Solution.   According to equation (5-20),   ,    where   ,  and   .  

Then we have      where     and  .  

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

We are done.   

Aside.  We can let Mathematica double check our work.

                    The mapping     and it's branch cut   .

This solution is complements of the authors.

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