Exercise 11.  Show what happens when    and    in Example 2.13.  

11 (a)  Find the image of the y-axis:   .   

Solution 11 (a).

See text and/or instructor's solution manual.

The transformation in Example 2.13 is    and usually maps vertical and horizontal lines onto portions of hyperbolas.  

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  .  

The right half plane given by    is mapped onto the region in the right half plane satisfying   

,   or      and lies to the right of   .  

This is the region between the lines    and    in the right half of the w plane.  

Hence, the y-axis is mapped onto the curve  .

Alternate solution.  

Use polar coordinates    in the z-plane and    in the  w-plane.  The mapping    can be written as  

        ,   

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

and we get the equations      and  .  

We know that the upper half-plane given by    is mapped onto the first quadrant.  

The positive y-axis,  ,  is mapped onto the ray,   , which is   .  

The negative y-axis,  ,  is mapped onto the ray,   , which is   .    

This solution is complements of the authors.

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