Exercise 10.  Show that      maps the upper half-plane      

onto the domain indicated in Figure 11.84.                     Figure 11.84.

Hint 1.  Set   ,  ,   and   ,  ,   respectively, and let   .    

Hint 2.  Use the change of variable      in the resulting integral.

Solution 10.

See text and/or instructor's solution manual.

Answer.   ,   for convenience set  .   

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  

                       

For convenience set  .   

Now integrate and get    

                    .  

The first integral is easy to get

                    .  
            

The second integral can be found using the change of variable

                      
            
make these substitutions in the integral and get

                      

Next use the substitution      and get

                    

Now combine this with the first integral and get

                    
                    
Therefore,  

                    .

We are done.   

Aside.  We can let Mathematica double check our work.

Use the identity    and write the solution using logarithms

Therefore,  

                    .  

We are really done.   

Aside.  For illustration purposes we can graph the mapping   .   

                    The mapping   .  

We are really really done.   

        Observe that the conditions     and  ,   are met.

The images of   ,   are   ,   respectively.

                    


                    
                    
                    
From calculus we have      so we will use      and write

                    

Aside.  For illustration purposes we can graph the mapping   .   

                    The mapping   .  

We are really really really done.   

Remark 1.1   If the computer algebra  Mathematica is used to perform the integration then the answer is

                      

which gives the correct result but uses the a specialized hypergeometric function.

Remark 1.2   Mathematica 7 will get the following formula for the integral

            

            

Remark 2.   If the computer algebra Maple is used to perform the integration then the answer is

                    .  

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

We are really really really really done.   

Aside.  For illustration purposes we can graph some of the other mappings.  

                    The mapping   .  

                    The mapping   .  

                    The mapping   .  

We are really really really really really done.   

Aside.   It is possible to expand the integrand      in the following form

                    .  

Then integrating each term on the right side yields

                      

Combining the above result with  produces the desired solution

                    

This solution is complements of the authors.

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