We can easily show that   maps the four points    onto  ,  respectively.  
However, the composition functions in Equation (11-36) must be considered in order to visualize the geometry involved.  First, the bilinear transformation    maps the region    onto the right half-plane  ,  and the points    are mapped onto  ,  respectively.  Second, the function    maps the right half plane onto the W plane slit along its negative real axis, and the points ,  are mapped onto  ,  respectively.  Then the bilinear transformation   maps the latter region onto the W plane slit along the portion of the real axis , and the points    are mapped onto  ,  respectively.  These three compositions are shown in Figure 11.61.  

Figure 11.61  The composition mappings for  .  

Exploration

(c) 2006 John H. Mathews, Russell W. Howell