Example 1.10.  If  [Graphics:Images/ComplexGeometryContinuedMod_gr_65.gif],  then  

            [Graphics:Images/ComplexGeometryContinuedMod_gr_66.gif]   and  

            [Graphics:Images/ComplexGeometryContinuedMod_gr_67.gif].  

It would be a mistake to use [Graphics:Images/ComplexGeometryContinuedMod_gr_68.gif] as an acceptable value for [Graphics:Images/ComplexGeometryContinuedMod_gr_69.gif], as the point z associated with [Graphics:Images/ComplexGeometryContinuedMod_gr_70.gif] and [Graphics:Images/ComplexGeometryContinuedMod_gr_71.gif] is in the first quadrant, whereas [Graphics:Images/ComplexGeometryContinuedMod_gr_72.gif] is in the third quadrant.  A correct choice for [Graphics:Images/ComplexGeometryContinuedMod_gr_73.gif] is [Graphics:Images/ComplexGeometryContinuedMod_gr_74.gif].  Thus

            [Graphics:Images/ComplexGeometryContinuedMod_gr_75.gif],  and

            [Graphics:Images/ComplexGeometryContinuedMod_gr_76.gif],
            
where n is any integer.  In this case,

            [Graphics:Images/ComplexGeometryContinuedMod_gr_77.gif],  and

            [Graphics:Images/ComplexGeometryContinuedMod_gr_78.gif].

Explore Solution 1.10.

[Graphics:../Images/ComplexGeometryContinuedMod_gr_83.gif]

 

 

 

[Graphics:../Images/ComplexGeometryContinuedMod_gr_84.gif]

[Graphics:../Images/ComplexGeometryContinuedMod_gr_85.gif]

 

 

 

Here is another way to look at it.

 

[Graphics:../Images/ComplexGeometryContinuedMod_gr_86.gif]

 

 

[Graphics:../Images/ComplexGeometryContinuedMod_gr_87.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell