Basic Theorems Regarding Harmonic Functions
 Proposition 1: Let $A \subseteq \mathbb{R}^2$ be open and let $f, g : A \to \mathbb{R}$ be harmonic on $A$. Then: a) $f + g$ is harmonic on $A$. b) $kf$ is harmonic on $A$ for all $k \in \mathbb{R}$.
• Proof of a) Suppose that $f$ and $g$ are harmonic on $A$. Then the second partial derivatives of $f$ and $g$ exist and moreover we have that $\Delta (f) = 0$ and $\Delta (g) = 0$. Then on all of $A$ we have that:
• So $f + g$ is harmonic on $A$.
• Proof of b) Suppose that $f$ is harmonic on $A$ and $k \in \mathbb{R}$. Then:
• So $kf$ is harmonic on $A$.