7. Proposition 2.1.1

Let $u (x, y)$ be a Harmonic function (then it is assumed also that $u$ is differentiable). Then there exists a $v (x, y)$ such that 1;; $f (x, y) := u (x, y) + i v (x, y)$ is holomorphic. Moreover, such a $v$ is unique up to a constant, and these other such $v$ are called conjugates of $v$ .

Remark

Rewrite this and mention Poincare Lemma. The idea is that we can find $v (x, y)$ locally by looking at an open ball around a point in our open set. I think we conclude this by looking at $\frac{δ u}{δ x} d y - \frac{δ u}{δ y} d x$ , and trying to argue that is a closed form and thus exact by Poincare.