Exercise 5.  Pick an appropriate branch of the logarithm (see Equation (5-20)), find  ,  and state where the formula is valid for  

5 (c).  .

Solution 5 (c).

See text and/or instructor's solution manual.

Solution.  Choose  ,  then    and      for    and   .

Then we have    which is the same as  

                       for    and   .

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

                              The image of the upper half-plane under the mapping  .  

                              The image of the lower half-plane under the mapping  .  

This solution is complements of the authors.

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