It is very possible that this question was asked before, but I cannot find it from the list.
Question. Let $f : [a, b] \rightarrow \mathbb{R}$ be infinitely differentiable on $(a, b)$. Can we extend this to a holomorphic function $F : D \subseteq \mathbb{C} \rightarrow \mathbb{C}$ where $D$ is an open disc such that $D \cap \mathbb{R} = (a, b)$?
Motivation to the question is simply that I want to organize the concept of Taylor expansion of real variable in terms of complex variable, and this question seems very plausible to me but not immediate.