Representation of harmonic function as a series

Question

Let, $u:D\to\mathbb{C}$ is a harmonic function then how to show that $u(z)=\sum_{n\in\mathbb{N}} a_n z^n + \sum_{n\in\mathbb{N}}b_n\bar{z}^n$. (D be the unit disk).

Answer 1

Let us quickly show that $u$ decomposes into a holomorphic and antiholomorphic function.
Assume that $u=f+ig$.
Since $f$ is harmonic and $D$ is simply connected, it has a harmonic conjugate $g_c$ such that $f+ig_c$ is holomorphic.
Furthermore, $g$ has also a harmonic conjugate such that $f_c+ig$ is holomorphic. Then the functions $$ F=\frac{f+f_c}{2}+i\frac{g+g_c}{2} \\ \bar{G}=\frac{f-f_c}{2}-i\frac{g_c-g}{2} $$ are holomorphic and antiholomorphic, respectively. Also we have $u=F+\bar{G}$ and a holomorphic function can be written as a power series in $z$, while an antiholomorphic function can be written as a power series in $\bar{z}$.