Exercise 11.  Show that      maps the upper half-plane      

onto the domain indicated in Figure 11.85.                     Figure 11.85.

Hint.  Set  ,  ,  and  ,  ,  respectively, and let   .    

Solution 11.

See text and/or instructor's solution manual.

Answer.   ,   integrate and get   ,  

then use the conditions      and obtain   .

Answer.    ,  integrate and get  .

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  

The exterior angles are  ,  and the formula for the derivative is  

                    

Integrate and get  

                    

        The images of   ,   are   ,   respectively.

Use   ,   and obtain the system of equations

                      

Which simplifies to be   

                      

The values    are solutions for this system of equations.

Therefore,   

                    .

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

        We can let Mathematica graph some of the streamlines.

                    The streamlines in the w-plane for the mapping   .   

We are really really done.   

Aside.  For illustration purposes we can graph the mapping   .  

                    The mapping   .    

This solution is complements of the authors.

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