Exercise 1.  Find the images of the mapping    in each case, and sketch the mapping.

1 (g).  The region the in the first quadrant between the hyperbolas  .

Solution 1 (g).

See text and/or instructor's solution manual.

Answer.  The infinite strip  ,  which is the region in the uv plane between  .  

Show the details in a manner similar to the answer for part 1(a).

Solution.   The mapping is    and

the point    in the xy-plane is mapped to the point    in the uv-plane,

and we get Equations (2-9);    and  .  

Use the equation  .  For points in the first quadrant region between the hyperbolas  ,  will satisfy  ,  

we see that the image points will satisfy  .  Therefore, the image is the horizontal strip  .  

We are done.   

Aside.  We can let Mathematica double check our work.

          The image of the region the in the first quadrant between the hyperbolas   under    is the infinite strip  .

This solution is complements of the authors.

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