Exercise 12.  Find the image of the upper half-plane      under   .  

Solution 12.

Answer.   The image of the upper half-plane      under   

is the horizontal strip      slit along the ray   .

Short Solution.   The image of      under      is   ,   then the image of  

  under    is the horizontal strip    slit along the ray  .

Solution.   The mapping    can be written as a composition  

                    ,    

where   

                    ,    and   

                    .

The upper half-plane      can be expressed in the form

                    .  

        The mapping   can be expressed as

                    ,  

                    or       and    .  

Then      implies      implies   .   

We can solve      and get   .  

Then      implies      implies      which in turn implies   .  

Thus, the image of        under        is    ,

which is the Z-plane slit along the ray   .  

        However, the mapping at hand is      and it shifts the above region  S  to the right by the amount  .

Thus, the image of      under      is   ,

which is the Z-plane slit along the ray   .  

        It helps if we determine the image of the set  .

Use       and the formulas

                        and    .   

Then      implies that     which in turn implies that     .  

Also,      implies that   .

        Hence, the image      under the mapping   

is the horizontal strip   .   However, the required image will be missing a certain ray.

        Recall that the domain of      is the Z-plane slit along the ray   .  

But our set  T  is the Z-plane slit along the ray   .    

So the image of  T  will be the horizontal strip   

minus the image of the interval   .

Here we have      implies that     which in turn implies that   .  

Also, we have      so that   .   

Thus, the image of the segment    under     

is the ray   .

        Hence, the image of the set      under the mapping     

is the horizontal strip      slit along the ray   .

        Therefore, the image of the upper half-plane      under   

is the horizontal strip      slit along the ray   .

We are done.   

          The mapping   ,   followed by the mapping   .

          Observe the points and their images    and    

                                                              

The image of the branch cut for    has four pieces which are shown with dashed lines and rays in the w-plane.  

The point    can be explained in more detail, and depends on which quadrant you approach  ,
this requires visualizing the four values of zero:.  

We are really done.   

Aside.   If it is important then we can include the following details.

          Here the values   ,    ,    ,    ,    ,        are calculated as limits.

                    ,  

                    ,  

                    ,  

                    ,  

                    ,  

                    .    

This solution is complements of the authors.

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