I have some problem that I can't figure out myself. Hope that someone can help me out. The problem is: Let $f : U \to \mathbb{R}$ be some real function on a simply-connected domain $U \subset \mathbb{C}^{n}$ with $\partial\bar{\partial}f = 0$ on $U$. Then $f$ is the real part of a holomorphic function on $U$.
The question is: Is this holomorphic function unique? If not what choices are there to choose such a function? Are there any choices? How does the proof go?
I hope that someone has some answers for me and I will be very happy for a lot of answers. I have to apologize if this question is too trivial for some of you :).
Best regards philippe