Definition:Analytic Function

Definition

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:

$(1): \quad \xi \in \left({c \,.\,.\, d}\right)$

$(2): \quad \displaystyle \forall x \in \left({c \,.\,.\, d}\right): f \left({x}\right) = \sum_{n \mathop = 0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({x}\right)$

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.

See Also

• Equivalence of Definitions for Analytic Function