Exercise 9.  Construct a branch of    that is analytic at the point    and takes on the value    there.  

Solution 9.

See text and/or instructor's solution manual.

Answer.   According to equation (5-20),   ,    

has  .

Solution.   Let   ,   where   ,  and    .  

Use equation (5-20),   ,   where   ,   and   .  

For    and  ,  and    we have    and  ,  i. e.  .

Then  .  

We are done.   

Aside.  We can let Mathematica double check our work.

                    The mapping     where the image of     is   .

This solution is complements of the authors.

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