Here, visIm(f(z)) so if it is constant, the right side of each of those two equations is 0 and we have and , the Cauchy-Riemann equations. In other words, that both u and v are constant.

Similarly, if Re(f(z)) is constant, so is u and then you can prove that v is constant.

Finally, if |f|= is constant, so is constant and we have and . Use those two equations together with the two Cauchy-Riemann equations to show that the four derivatives , , , and are all 0.