Exercise 12.  Show that      maps the upper half-plane      

onto a right triangle with angles  ,  ,  and  .

Hint.  For convenience we have chosen  .  We will permit   to be unknown but fixed values.

Solution 12.

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 ,  .  

The solution can be left in the form of an integral.    

Therefore,   

                    .    

We are done.   

        Formulas for integrals such as      are not part of our course.  

They involve special functions such as   Hypergeometric2F1.  

We are really done.   

The following graphs are included to encourage you to research this topic.  

                    The mapping   .    

                    The mapping   .    

                    Here the unit disk has been mapped onto the upper half plane with  .

We are really done.   

Aside.  We can see if the computer software Mathematica can find the integral.

Remark 1.   Mathematica 7 will get the following formula for the integral

            
            
            

Remark 2.   The function   Hypergeometric2F1   is not part of our course.  

These integrals have been to encourage you to research this topic.  

This solution is complements of the authors.

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