Definition:Analytic Function/Real Numbers

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Definition

Let $f$ be a real function which is smooth on the open interval $\openint a b$.

Let $\xi \in \openint a b$.


Let $\openint c d \subseteq \openint a b$ be an open interval such that:

$(1): \quad \xi \in \openint c d$
$(2): \quad \displaystyle \forall x \in \openint c d: \map f x = \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } x$


Then $f$ is described as being analytic at the point $\xi$.


That is, a function is analytic at a point if it equals its Taylor series expansion in some interval containing that point.


Also known as

This is also known as a real analytic function.


Sources