Analytic in a Domain D
if f(z) is analytic/holomorphic in a Domain D, prove f(z) must be constant in that domain if any of these are also constant in D:
Im(f(z)), Re(f(z)), |f(z)|
So, f(x + iy) = u(x, y) + iv(x, y)
u, and v have first partial derivatives and satisfy Cauchy-Riemann equations:
How can you go about showing this? Thanks.
Here, v is Im(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.
Originally Posted by universalsandbox
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.