Exercise 5.  Find all values of  z  for which the following equations hold.  

5 (d).  .

Solution 5 (d).

See text and/or instructor's solution manual.

Answer.   ,   where n is an integer.

Solution Method I.   Use the relation  ,  and observe that

                       and   .

Using polar coordinates    we have    and    calculate  

                       and   .

Thus,   ,   where n is an integer.

Solution Method II.   Start with

                      

Then,   ,   where n is an integer.

Solution Method III.   Use equation (5-9)  .  

Start with  ,  and compute    and  ,  and then  

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

                                                            Solutions to the equation   .  Where the image point is  ,   
                                                            and the principal solution value is  ,   
                                                            and some of the points  ,   where n is an integer. 

This solution is complements of the authors. 

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