Definition:Analytic Function

Let $$f$$ be a real function which is smooth on the open interval $$\left({a \, . \, . \, b}\right)$$.

Let $$\xi \in \left({a \, . \, . \, b}\right)$$.

Let $$\left({c \, . \, . \, d}\right) \subseteq \left({a \, . \, . \, b}\right)$$ be an open interval such that:


 * $$\xi \in \left({c \, . \, . \, d}\right)$$;
 * $$\forall x \in \left({c \, . \, . \, d}\right): f \left({x}\right) = \sum_{n=0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({x}\right)$$.

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.