Question

Assume that the following result is true: If u(x, y) has

partial derivatives ux and uy and ux(x, y) = uy(x, y) for every point
z = x + iy in a region Ω, then u is constant in Ω.
Use this to prove the following result:
Let f = u + iv be analytic in a region Ω. If u or | f | is constant in Ω,
then f is constant.

Answer 1

We are given a result: If partial derivatives of the real part u are equal at every point in a region Ω, then u is constant in Ω.

Now, let's prove: If u or | f | is constant in Ω, then the complex function f = u + iv is constant in Ω.

Proof:

1. If u is constant: If u is constant, both partial derivatives of u are zero. By the given result, u is constant in Ω.
2. If | f | is constant: If | f |^2 = u^2 + v^2 is constant, both u^2 and v^2 are constant. This satisfies the given result's conditions for u and v. Using the Cauchy-Riemann equations, we find that both u and v are constant. Therefore, f = u + iv is constant in Ω.

Hence, if u or | f | is constant in Ω, then f = u + iv is also constant in Ω.