Exercise 1.  Let    and    be real constants with  .

Use the Schwarz-Christoffel formula to show that the function     

maps the upper half-plane      onto the sector   ,   

as shown in Figure 11.75.                     Figure 11.75.

Solution 1.

See text and/or instructor's solution manual.

Answer.   ,  integrate and get  .   

Then choose     and obtain   .  

Solution.   Set  ,  and  ,  respectively,

The exterior angle is  ,  and the formula for the derivative is  

               

Integrate and get

            
            
Use the condition    and (and choose    for convenience)  and  obtain the desired result.

Therefore,

            .

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 mapping   .   

This solution is complements of the authors.

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