(a) Consider an open $\operatorname{disc} D \subseteq \mathbb{C}$. Prove that a real-valued function $u: D \rightarrow \mathbb{R}$ is harmonic if and only if

$$u=\operatorname{Re}(f)$$

for some analytic function $f$.

(b) Give an example of a domain $D$ and a harmonic function $u: D \rightarrow \mathbb{R}$ that is not equal to the real part of an analytic function on $D$. Justify your answer carefully.

(c) Let $u$ be a harmonic function on $\mathbb{C}_{*}$ such that $u(2 z)=u(z)$ for every $z \in \mathbb{C}_{*}$. Prove that $u$ is constant, justifying your answer carefully. Exhibit a countable subset $S \subseteq \mathbb{C}_{*}$ and a non-constant harmonic function $u$ on $\mathbb{C}_{*} \backslash S$ such that for all $z \in \mathbb{C}_{*} \backslash S$ we have $2 z \in \mathbb{C}_{*} \backslash S$ and $u(2 z)=u(z)$.

(d) Prove that every non-constant harmonic function $u: \mathbb{C} \rightarrow \mathbb{R}$ is surjective.