Pseudoanalytic function

Remarks

The condition $\operatorname{Im}(\bar F G) > 0$ says that $F(z)$ and $G(z)$ are linearly independent over $\R$, which gives the uniqueness of $\varphi$ and $\psi$.

Example: for $(F, G) = (1, i)$, the condition reads $(\varphi + i \psi)_{\bar z} = 0$. So the $(1, i)$-pseudoanalytic functions are exactly the holomorphic functions with continuous derivative.

Non-example: $w(z) = \bar z$ is not $(1, i)$-pseudoanalytic. Here $\varphi = x$ and $\psi = -y$, so $\varphi_{\bar z} + i \psi_{\bar z} = \frac{1}{2} + i \cdot \left( -\frac{i}{2} \right) = 1 \ne 0$.

If $F$ and $G$ are continuously differentiable, then $w_{\bar z} = \varphi F_{\bar z} + \psi G_{\bar z}$ for a pseudoanalytic $w$. Since $\varphi$ and $\psi$ are real-linear in $w$, this is an equation $w_{\bar z} = a w + b \bar w$ with coefficients $a, b$ determined by $F$ and $G$. Some authors ask for less regularity of $\varphi$ and $\psi$ and use a weak form of the condition.