Exercise 9.  Describe the set of complex numbers for which  .  Prove your assertion.  

Solution 9.

See text and/or instructor's solution manual.

    Answer:  The negative real numbers and the number     .  Prove this!

    Assume that z is a complex number that is not a negative real number or zero.   Then and , and z can be represented in polar coordinates as

          where  .  Then    and  .  

This proves that    when z is not a negative real number or zero.

    If     .  then ,   , and are all undefined.  So   by default.

If  z  is a negative real number then  ,  where .  Calculation reveals that  ,  and  .  In this case we see that  

        ,  
and
        .

Therefore,    when  z  is a negative real number.

This solution is complements of the authors.

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