# Extension of real smooth function to a holomorphic function.

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.

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I implicitly meant $F|_{(a, b)} = f$ although it is not included. – user123454321 Oct 31 '12 at 16:06
Later realization: I was confused between when we can use Taylor's theorem and have analytic function. The hypothesis given in the question is enough for us to use Taylor series but we cannot expand it in an infinite sum because our remainder is too large. Old John's answer is a good example to see this. I might ask a new independent question as I was interested in expanding analytic function. – user123454321 Oct 31 '12 at 18:22

You might want to consider the function $f(x)=e^{-1/x^2}$, which is infinitely differentiable as a real function. All the derivatives are zero at the origin, so it is not possible to get a Taylor series for the function which represents the function away from zero, and as a complex function, it has an essential singularity at the origin, so is not holomorphic in any region which includes the origin.
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