Exercise 5.  Find the image of the right half-plane    under the mapping  .  

Solution 5.

See text and/or instructor's solution manual.

Answer.  The region in the w-plane that lies to the right of the parabola  .  

Solution.  The mapping    is the composition of  the functions    and  ,  i. e.  

        .  

The function    maps the right half-plane    onto the right half-plane  .  

Now follow the steps in 1 (b).   

For the mapping    the point    in the XY-plane

is mapped to the point    in the uv-plane,  and we get the equations    and  .  

Then, using  ,  the first quadrant    is mapped onto the upper half-plane,

so the image of    is contained in the upper half plane  .  

From Exercise 1 (b), the image of the line    under the mapping    is the parabola  .

Hence, the image of the right half plane     under the mapping    

is region in the w-plane that lies to the right of the parabola  .  

Therefore, the the image of the right half-plane    under the mapping  

is the region in the w-plane that lies to the right of the parabola  .  

We are done.   

Aside.  We can let Mathematica double check our work.

                    The image of    under the mapping    lies to the right of the parabola  .

This solution is complements of the authors.

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