I am trying to solve a problem:
Let $f(z)$ be a smooth real-valued harmonic function on the punctured unit disc $\mathbb{D}^* = \mathbb{D} \setminus \{0\}$. Show that \begin{align} f(z) = \mathrm{Re}(F(z)) + c \log|z| \end{align} where $F(z)$ is a holomorphic function on $\mathbb{D}^*$ and $c \in \mathbb{R}$.
What I did: Let $H = \{x+iy \in \mathbb{C}: x<0\}$ and consider $g(z) = e^z$. Then, $g$ is holomorphic in $H$ and it sends $H$ to $\mathbb{D}^*$. Since $H$ is simply connected, $f\circ g$ is harmonic on $H$ and there exists a harmonic conjugate $v$ such that $f\circ g + iv$ is holomorphic in $H$. Using C-R equation and $2\pi i$ periodicity of $f\circ g$, we have $v(z + 2\pi i) = v(z) + C$ for some constant $C$. Define $G(z) = f \circ g + iv - \frac{C}{2\pi}z$. Then, $G$ is holomorphic in $H$ and $2\pi i$-periodic.
I'd like to claim that $G(\log z)$ is well-defined in $\mathbb{D}^*$, but I am not sure it is holomorphic in $\mathbb{D}^*$. Is it holomorphic? If so, how to show it?