Solution 4.

See text and/or instructor's solution manual.

Solution.     For a given angle  ,  the line    inclined at the angle    consists of two rays in the  z-plane   

                    ,    and  
                    
                    .  

Using the polar coordinates    and  

for the mapping      we have      and     

                    .  

Thus the images of    and    are  

                    ,    and  
                    
                    .  

Observe carefully that

                      

Hence the image of the two rays    and    in the  z-plane  is the single ray    in the  w-plane.   

Therefore,  the image of a line    inclined at the angle    is the ray    inclined at the angle  .

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  We can explore some graphs.

                    The line    formed by two rays    and  .  

                    The images of      are the rays        and    .  

This solution is complements of the authors.

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