Definition:Analytic Function

On the Reals
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.

In the Complex Plane
Let $$f \left({z}\right): \mathbb{C} \to \mathbb{C} \ $$ be a single-valued continuous complex function in a domain $$D \subseteq \mathbb{C}$$.

Let $$f$$ be complex-differentiable in $D \ $.

Then $$f \left({z}\right) \ $$ is described as being analytic on $$D$$.

An analytic complex function is also referred to as a holomorphic function.

A function which is analytic except at a countable number of isolated points is called meromorphic.