Let f : ℂ → ℂ be a real-differentiable function. Define u, v : ℝ2 → ℝ by u(x, y) = Re f(x + i y) and v(x, y) = Im f(x + iy), x, y ∈ ℝ.
Let ∇u = (ux, uy) denote the gradient. Which one of the following is necessarily true?
1
For c1, c2 ∈ ℂ, the level curves u = c1 and v = c2 are orthogonal wherever they intersect.
2
∇u . ∇v = 0 at every point.
3
If f is an entire function, then ∇u . ∇v = 0 at every point.
4
If ∇u . ∇v = 0 at every point, then f is an entire function.