Example 11.29.  Show that the mapping    maps the upper half plane   onto the domain in the w-plane that lies above the boundary curve consisting of the rays   and the segment     (see Figure 11.87).

            Figure 11.87  (a) Flow over a step.                  (b) Flow around a blunt object.

Explore Solution 11.29.

Use the Schwarz Christoffel formula, enter the formula    and integrate it to construct  f(z).

This is one, formula for the integral.   However, we will use the following form of the integral to continue the computations.

Now solve for the coefficients  A  and  B.

Unfortunately, the branch cuts don't work out right and we will have to do this one by hand.

Using this formula in the book we form the two branches of the function.  

Use Mathematica to graph conformal mapping  w = f(z).

We have shown that the mapping    maps the upper half plane   onto the domain in the w-plane that lies above the boundary curve consisting of the rays   and the segment  .    

We are done.

Aside.  Here is another function that produces the desired result.  The graph is drawn parametrically in this case.

Also, we have shown that the mapping    maps the upper half plane   onto the domain in the w-plane that lies above the boundary curve consisting of the rays   and the segment  .    

(c) 2006 John H. Mathews, Russell W. Howell