Ok, so let us suppose that functions u(x,y) and v(x,y) are harmonic in a domain D. Thus uxx+uyy=0, and vxx+vyy=0. In addition, v is a harmonic conjugate to u. Thus ux=vy, and uy=-vx. *Note ux= partial derivative of u with respect to x, vx=partial derivative of v with respect to x, and so on. I have to use all of this information to show that functions u and v must be constant throughout domain D.