Decomposition of Harmonic function into sum of holomorphic and anti-holomorphic function

Question

How do you prove that a harmonic planar mapping $f(x,y) = u(x,y) + i v(x,y)$ for real $u,v$ can be written as $f(x,y) = \phi(x,y) + \overline{\psi}(x,y)$ where $\phi$ is a holomorphic function, and $\overline{\psi}$ is an anti-holomorphic function (conjugate of a holomorphic function)?

Answer 1

Since $u$ is harmonic, it has a harmonic conjugate $\hat{v}$ such that $u+i\hat{v}$ satisfies the Cauchy-Riemann equations. Similarly $v$ has a harmonic conjugate $\hat{u}$ such that $\hat{u}+iv$ satisfies the Cauchy-Riemann equations.

Let $\phi=\frac{u+\hat{u}}{2}+i\frac{v+\hat{v}}{2}$ and $\psi=\frac{u-\hat{u}}{2}+i\frac{\hat{v}-v}{2}$. Then $\phi$ and $\psi$ are holomorphic, and $f=\phi+\overline{\psi}$.