Exercise 8.  Show that      maps the upper half-plane      

onto the domain indicated in Figure 11.82.                     Figure 11.82.

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

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

Solution 8.

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 ,  .  

Integrate and get

                    .  

The first integral is easy to get

                    .

The second integral can be found using the suggested change of variable  
            
                      

Make substitutions in the integral

                      

Now use the substitution      and get  

                    .

Now combine this with the first integral and obtain  

                    
        
Therefore,   

                    .

We are done.   

Aside.  We can let Mathematica double check our work.

The logarithm term could also be written in the form  

                     .

We are really done.   

Aside.  We can check some more work.

To get the desired formula replace    with  .

We are really really done.   

Aside.  For illustration purposes we can graph the mapping   .  

                    The mapping   .  

We are really 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

                    

Remark 1.   And if the inverse hyperbolic functions are used then the solution can be written as

                     .

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

                    .  

Remark 3.    If the computer algebra Maple is used to perform the integration then the answer is

                    .  

Or if the second integral is treated separately, then Maple's answer will be  

                    .  

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

We are really really really really done.   

                    The mapping   .  

                    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

                      

This solution is complements of the authors.

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