Hallo,

I have the following PDE that I am trying to solve via the Cauchy-Kowalewski Theorem. But I have no idea how to do it or if its possible. Maybe one of you has an idea. Here is the problem: Let $U \subset \mathbb{C}^{n}$ be some open subset witch contains zero. Well you can shrink $U$ arbitralily if you wish. Let $z_{j} = x_{j} + i y_{j}$ be the coordinates on $U$. I am looking for a real-valued analytic function $\beta$ defined on $U$ such that the equations are satisfied: $\frac{\partial \beta}{\partial x_{j}} = F_{j}(x,y,\beta)$ and $\frac{\partial \beta}{\partial y_{j}} = G_{j}(x,y,\beta)$,with $j = 1, ..., n$, where $F,G$ are also real analytic functions and with initial condition $\beta(x,0) = 0$, $\forall x \in U \cap \mathbb{R}^{n}$. Is it possible to solve such an system of equations via Cauchy-Kowalewski ? If yes, how? If no, why and is there any other method that can give me a solution? I hope that for a lot of answers and please excuse me if the question is too trivial. Thanks in advance.

Greeting Andrei