Finding all functions making $u + iu$ holomorphic

Question

I wish to find all possible $C^{1}$ functions $u: \mathbb{R}^2 \rightarrow \mathbb{R} $ s.t. $f(x + iy) = u(x,y) + iu(x,y)$ is analytic/holomorphic/complex differentiable. This would occur if and only if $u$ should satisfy the Cauchy-Riemann equations $\partial_xu = \partial_y u, \partial_y u = -\partial_x u.$ Am I then correct in deducing that the only functions $u$ satisfying these C-R equations would be constant functions (i.e. $\nabla u = 0$)? Have I missed any other?

Answer 1

You're right that $u$ needs to be a constant function. Another way to see it is to notice that $f(x,y)= u(x,y) (1+ i)$ and that a non-constant holomorphic map is open.

Note: I don't know what you mean by a constant map whose $\nabla u = 0$ as $\nabla u = 0$ is always true for a constant map.