Exercise 9.  If c is a complex number, the expression    may be multivalued.  

Suppose all the values of    are identical.  

What are these values, and what can be said about the number   ?  Justify your assertions.  

Solution 9.

See text and/or instructor's solution manual.

Solution.  Write  c  in the complex form  .  Then  

                       

Taking the absolute value we have

                    .

The principal value occurs when n=0 and is  .  

Since all these values are equal we must have    which implies that   which in turn implies that .

Therefore,  ,  and we must have     .  

Therefore  c  must be a real number.  

We are done.   

Aside.  We can let Mathematica double check our work.

This solution is complements of the authors.

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