Solution 5.

See text and/or instructor's solution manual.

Answer.   .   

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  

               

Now integrate
            

The integral can be found using the suggested change of variable  

            
            
            

Make substitutions in the integral

              

Now use the substitution  ,  

            

            

            

Solve for  ,  and  ,  and get the following solution

            .  

The first logarithm term can be rewritten in the following forms  

            ,
            
            .         

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

        We can let Mathematica graph some of the streamlines.

        The flow up an inclined ramp, as shown in Figure 11.92(a).   

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

                    ,    for    .  

We are really really done.   

Aside.  We can extend the flow into the third and fourth quadrants using symmetry.

        The flow around a pointed object, as shown in Figure 11.92(b).   

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

                    ,    for    .  

We are really really really done.   

Aside.  We can use Mathematica to graph   

                    .  

                    The image of the upper half plane    under the mapping   

                    .  

We are really really really really done.   

Aside.  We can extend the mapping into the third and fourth quadrants using symmetry.

                    The image of the complex plane under the mapping   

                    .  

Aside.  We can graph the other version involving logarithms

                    .  

                    The image of the upper half plane    under the mapping   

                    .  

We are really really really really really done.   

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

            ,   

which gives the correct result but uses the a specialized hypergeometric function.

Aside.  Here are two extra graphs using two different mappings:

                       and  

                    .

                    The image of the upper half plane    under the mapping   

                    .  

                    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