My book goes through this, but there are some things that I don't understand. Let z = h(w), where h is analytic. Then F(w) = f(h(w)) is also analytic if f(z) is analytic. Thus in the z and w plane f and F satisfy the Laplace equation. My book then says: "Further, suppose that Re f(z) is constant over a boundary C in the z-plane, then Re F(w) is constant over the boundary C' in the w-plane, where C' is the curve that C is transformed to in the w-plane.." I have emphasized the last bit, because this is the thing I don't understand. What allowed you to make the conclusion that Re F(w) is constant over the boundary C' in the w-plane? Certainly the Re F(w) can also be a function of Im f(z), right?