Exercise 15.  Show that the transformation      is a one-to-one conformal mapping of

the semi-infinite strip      onto the upper half plane  .

Solution 15.

Hint.   Extend the results in Example 10.13.  You can use the information in Figure 10.17.

Short Solution.   The image of the semi-infinite strip    under    

is the first quadrant   .   Then the image of the first quadrant      

under      is the upper half-plane   .  

Therefore, the image of the semi-infinite strip      under the mapping      

is the upper half-plane   .  

We might be done.   

Solution.   The mapping    can be written as a composition  

                     ,    

where   

                    ,    and   

                    .

        Using Equation (5-33), we write  

                    .  

If  ,  then the image of the vertical line    is the curve in the Z-plane given by the parametric equations  

                        and   ,    for    .  

The inequalities  ,  and    imply that the image points lie in the first quadrant  .  

Eliminate y from the parametric equations and the result is  .  

When the image curve approaches the positive Y axis where  .  

When the image curve approaches the ray  .  

        Hence, the image of the semi-infinite strip      under the mapping     

is the first quadrant  .  

        The first quadrant    can be expressed as

                    .  

The mapping    can be written as  

                    ,    where  

                        and    .  

Then      implies      which in turn implies that   .  

Also,      implies      which in turn implies that   .  

        Hence, the image of the first quadrant    under      is the upper half-plane    

                    .  

        Therefore, image of the semi-infinite strip      under the mapping     

is the upper half-plane   .  

We are done.   

Aside.  We can look at some graphs of the mapping  .

          The mapping   ,   followed by the mapping   .

          Observe the points and their images    and    

                                           

This solution is complements of the authors.

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