Solution 1.

See text and/or instructor's solution manual.

Answer.   The transformation  ,  of Example 11.27 in Section 11.9, is known to map a horizontal flow in the

upper half plane    onto flow the upper half plane    slit along the vertical line segment from  .  

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

                    ,     for     .  

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

The exterior angles are   ,   respectively,

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

                      

Now integrate and get

                      

        The images of   ,   are   ,   respectively.

Use     and obtain thel system of equations

                       and   

the solution is easily found to be  .

Therefore,  

                    .

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    .  

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    using a conformal branch of   .

We are really really really really done.   

Aside.  We can use a different formula for the branch of the square root.

                    The image of the upper half plane    using a conformal branch of   .

This solution is complements of the authors.

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