I have a two part question about real analyticity and whether I have a an error in reasoning.
Suppose I have a multivariate holomorphic mapping $f \colon U \to \mathbb{C}$, where $U$ is an open set in $\mathbb{C}^n$, with $f(U \cap \mathbb{R}^n) \subset \mathbb{R}$ would $f|_{U \cap \mathbb{R}^n}$ then be real analytic?
My thinking is, that I can show, that all the coefficients in the power series expansion must be real because I can calculate the limit of the difference quotients over the real numbers. I have seen such an argument in the univariate case in a different post but am not sure whether there are intricacies in the multivariate case.
If this is correct would the inverse/implicit function theorem for real analytic mappings directly follow from the version for holomorphic mappings? I.e. for the inverse function theorem and a real analytic mapping $f \colon U \to \mathbb{R}^n$ with regular $df(z)$, $z \in U$, take the complex candidate of $f^{-1}$ but how can one show, that $f^{-1}$ is real on the real numbers?
Thank you!