Exercise 5.  Show that      maps the upper half-plane      

onto the infinite strip      slit along the ray   ,  

as shown in Figure 11.79.                     Figure 11.79.

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

Solution 5.

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.

Use    and  ,  and obtain the system of equations

                    

Which simplifies to be   

                    

From calculus we have      so we will use      and write

                    

The values    are solutions for this system of equations.

Therefore,   

                    .

Or if you prefer, you can write the solution as  

                    .

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  For illustration purposes we can graph the mapping   .  

                    The image of the upper half plane    under the mapping   .  

This solution is complements of the authors.

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