Exercise 4.  Show that      maps the upper half-plane

onto the domain indicated in Figure 11.78.                     Figure 11.78.

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

Solution 4.

See text and/or instructor's solution manual.

Answer.   ,   integrate and get   ,  

then use the conditions         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.

Use   ,   and obtain the system of equations  

                    

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   .  

We are really done.   

If you prefer, the square root can be written in the following form

                    .

If you prefer, the trigonometric function can be written in the form

                    .

Aside.  We can let Mathematica double check our work.

We are really really done.   

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

                    The image of the upper half plane    under the mapping   .  

                    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