Find a harmonic function u in the open first quadrant that extends continuously up to the boundary except at the points 0 and 1, and that takes on the following boundary values: u(x, y) = 1 on the half-lines {y = 0, x> 1} and {x = 0, y> 0}, and u(x, y) = 0 on the segment.
[Hint: Find conformal maps F1, F2,...,F5 indicated in Figure. Note that 1/Π arg(z) is harmonic on the upper half-plane, equals 0 on the positive real axis, and 1 on the negative real axis.]
