476 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS
Figure 11.56- Show that F (z) = sin z is the complex potential for the ideal fluid flow inside the
semi-infinite strip -; < x < ~,y > 0 indicated in Figure 11 .57. Find the stream
function.
Figure 11.57- Let w = S (z) = ~ [z + (z^2 - 4)! ] denote the branch of the inverse of z = w +;
tba.t is a one-tcrone mapping of the z plane slit along the segment -2 $ x $ 2, y =
0 onto the domain lwl > 1. Use the complex potential F 2 (w) =we- "' + ·~" in
the w plane to show that the complex potential F 1 ( z) = z cos a -i ( z^2 - 4) i sin a
determines the ideal fluid flow around the segment - 2 $ x $ 2, y = 0, where t he
velocity at points distant from the origin is given by V ::::: eio<, as shown in Figure
11.58.