Exercise 6.  Show that      maps the upper half-plane      

onto the domain indicated in Figure 11.80.                     Figure 11.80.

Remark.  This solution can also be written in the form     

Hint.  Set   ,  ,   and   ,  ,   respectively.  

Solution 6.

See text and/or instructor's solution manual.

Answer.   ,   integrate and get   ,  

then use the conditions     and     and obtain   .

Solution.   Along the x-axis use the points   .   The exterior angles are  ,  

and the formula for the derivative is  given by the Schwarz-Christoffel formula  

                        

Integrate and get

                      

The images of   ,   are   ,   respectively.

Using   

with    and  ,  we obtain the system of equations

                           

Which simplifies to be   
                    
                           
                    
Then
                           

The values      are solutions for this system of equations.

Therefore,   

                    .

We are done.   

Aside.  We can let Mathematica double check our work.

Aside.  For illustration purposes we can graph the mapping   .  

                    The image of the upper half plane    under a conformal branch of

                    the mapping   .  

Alternative Solution.    Integrate and get

                      

The images of   ,   are   ,   respectively.

Using   

with     and  ,   we obtain the system of equations

                           

Which simplifies to be   
                    
                           
                    
Then

                           

The values      are solutions for this system of equations.

Therefore,   

                    .

We are really done.   

Aside.  We can let Mathematica double check our work.

We are really really done.   

Summary of Results.   The following two mapping of the upper half-plane will produce the same results.

                    The image of the upper half plane    under a conformal branch of

                    the mapping   .  

This solution is complements of the authors.

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