Does there exist an implicit function theorem (IFT) covering the following setting:
Consider $f\colon \mathbb{C} \times V \to V$ where $V$ is a Fréchet space, satisfying certain conditions. I wish to implicitly define $x\colon U \subset \mathbb{C} \to V$, where $U$ is an open neighborhood, such that $f(z, x(z)) = 0$ for all $z \in U$, and additionally conclude that $x$ is analytic.