Exercise 4.  Find and illustrate the images of the following sets under the mapping  .  

4 (e).  The infinite strip  .  

Solution 4 (e).

See text and/or instructor's solution manual.

Answer.  The region in the first quadrant that is bounded by the hyperbolas    and  .

Solution.   Given  ,  the inverse mapping is    and

the point    in the uv-plane corresponds to the point    in the xy-plane,

and we get the equations    and  .  Now substitute and then into the second equation.  

The horizontal line    is mapped onto the portion of the hyperbola    that lies in the first quadrant.

The horizontal line    is mapped onto the portion of the hyperbola    that lies in the first quadrant.

Therefore, the image of    is the region in the first quadrant that is bounded by the hyperbolas    and  .

We are done.   

Aside.  We can let Mathematica double check our work.

                    The image of the infinite strip    under the mapping    is
                    the region in the first quadrant that is bounded by the hyperbolas    and  .

This solution is complements of the authors.

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