Exercise 1.  Find    for the following values of  z.  

1 (d).  .  

Solution 1 (d).

See text and/or instructor's solution manual.

One solution is obtained by multiplying out the quantity   as follows:

        

then

Warning.  We obtain the calculated value    which is not the argument for a complex number in Quadrant III.    

To obtain the correct value of the argument we must subtract  ,  i.e.  .

For this example we need to use the formula for ArcTan that has two arguments (no pun intended) : .

We are done.   

    Another solution is obtained by using the formula  ,  where n is an integer.

In our case, this becomes    ,  where n is an integer.

        .  

If we choose  ,  then    lies in the interval,  

so the principal value of the argument is found to be

        

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