Exercise 15.  Show that      maps the upper half-plane      

onto the domain indicated in Figure 11.86.                     Figure 11.86.

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

Solution 15.

See text and/or instructor's solution manual.

Answer.   ,   integrate and get   ,  

The desired solution has  ,  .   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 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  ,  

                    .

A Solution.  Now combine this with the first integral and get  

                      

The desired solution has  ,  .  

Therefore,   

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

Remark.  To obtain our form of the function value  f(1)  use the computation   

                      

The Solution.  Use the version     with the first integral and get  

                      

The desired solution has  ,  .  

Therefore,   

                    .  

We are really done.   

Aside.  For illustration purposes we can graph the mapping   .  

                    The mapping   .  

We are really really done.   

Aside.  We can let Mathematica double check our work.

        Observe that the conditions   ,   are met.

The images of   ,   are   ,   respectively.

                    

and

                      

Remark.  To obtain our form of the function value  f(1)  use the computation   

                    .

We are really really really done.   

Another Alternative Solution.   The images of   ,   are   ,   respectively.

Using   

with   ,   we obtain the system of equations

                      

Then

                      

Then

                      
                    
Then

                      

The values    are solutions for this system of equations.

Therefore,   

                    .  

We are really really really really done.   

Aside.  For illustration purposes we can graph the mapping   .  

                    The mapping   .  

We are really really really really really done.   

Aside.  We can let Mathematica double check our work.

The logarithm term could also be written in the form  

                    .  

And if the inverse hyperbolic functions are used then this can be written as

                    .  

Remark 1.   If the computer algebra Mathematica is used to perform the integration then the answer is

                    .  

Remark 2.   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 really really done.   

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

                    The mapping   .  

                    The mapping   .  

                    The mapping   .  

                    The mapping   .  

We are really really 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