Exercise 13.  Find the branch of      that maps the right half-plane     

onto the right half-plane        slit along the segment   .  

Solution 13.

Answer.   Use  the principal branch of the square root  .

Solution.   The desired mapping  is    and can be written as a composition  

                    ,    

where   

                    ,   and   

                    ,  

and the principal branch of the square root    is used.   

        The right half-plane can be expressed as

                      

The mapping   can be expressed as

                    ,  

                    or       and    .  

Then      implies      implies   .   Here we have   .  

Then      implies      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  .

        Hence, the image of      under      is   ,

which is the Z-plane slit along the ray   .  

        It helps if we observe that the image of the set     under   

is the right half-plane .   However the required image will be missing a certain segment.

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 right half-plane

minus the image of the segment   .

Then      implies     implies     which in turn implies that   .  

Also, we have      so that   .   

Thus, the image of the segment    under     

is the segment   .

        Hence, the image of the set      under the mapping     

is the right half-plane        slit along the segment   .  

        Therefore, the image of the right half-plane        under    

is the right half-plane        slit along the segment   .  

We are done.   

          The mapping   ,   followed by the mapping   .

     It is easy to trace the images of      under the mapping  
which are mapped onto the points   .  

          Observe the points and their images    and    

                                                              

We are really done.   

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

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

Notice that for   we chose    and  for   we chose  .

This requires visualizing the two values of  :,
and depends on which half-plane you approach  ,  i.e.

                    ,    and  
                    
                    .  

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

                    ,    and  
                    
                    .  

          Observe the points and their images    and    

                                

This solution is complements of the authors.

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