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:


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

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

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.