I have a potential function $F(x,y)$ that satisfies the 2D Laplacian $\nabla^2 F = \frac{\partial^2 F}{\partial x^2}+\frac{\partial^2 F}{\partial y^2} =0$. Given the property of potential functions, a linear combination of complex functions in variable $z = x+iy$ and its conjugate $\bar{z} = x-iy$ must always satisfy the 2D Laplacian. Let these functions be $f(z)$ and $g(\bar{z})$.
i.e. we have $\nabla^2 (a f(z) + b g(\bar{z}) =0 $ for any set of complex constants $a$ and $b$. If so, given an $F(x,y)$ is there a way to find the complex functions $f$ and $g$?
For eg: If $F(x,y)=2 \cosh(x)\cos(y)$, I already know that my $f(z) =\cosh(z)$ and $g(\bar{z}) = \cosh(\bar{z})$, and $a=1, b=1$. Several such examples can be constructed by choosing $f$ and $g$. However, is there a way to find $f$ and $g$ from $F$?