Solution 2.

See text and/or instructor's solution manual.

Answer.   The transformation  ,  of Exercise 6 in Section 11.9,   

is known to map a horizontal flow in the upper half plane    onto flow the upper half plane    

that moves around the exterior of the infinitely long rectangular barrier.  

Solution.   Along the x-axis use the points      and in the w-plane use      respectively.    

The exterior angles are   ,  

and the formula for the derivative is  given by the Schwarz-Christoffel formula  

                

Integrate and get

                      

The images of   ,   are   ,   respectively.

Use    and  ,  and obtain thel system of equations

                           

Which simplifies to yield
                    
                    

The solution to this system of equations is easily found to be   .

Therefore,   

                    .

Or if you desire, you can use the equivalent formula

                    .  

We are done.   

Aside.   The image of horizontal streamlines in the z plane are curves in the w plane given by the parametric equation

                    

for  .  Or if you desire,

                    

for  .  

We are really done.   

Aside.  We can let Mathematica double check our work.

We are really really done.   

        We can let Mathematica graph some of the streamlines.

                    The streamlines in the w-plane given by the parametric equation

                    ,    for    .  

We are really really really done.   

Aside.  We can let Mathematica graph   .  

                    The image of the upper half plane    under the mapping   .

We are really really really really done.   

Aside.  We can replace    with    in the formula for  .

                    The image of the upper half plane    under the mapping   .

This solution is complements of the authors.

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