Definition:Analytic Function

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:

$(1): \quad \xi \in \openint c d$

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

Then $f$ is an analytic (real) function at the point $\xi$.

Let $U \subset \C$ be an open set.

Let $f : U \to \C$ be a complex function.

Then $f$ is analytic in $U$ if and only if for every $z_0 \in U$ there exists a sequence $\sequence {a_n}: \N \to \C$ such that the series:

$\ds \sum_{n \mathop = 0}^\infty a_n \paren {z - z_0}^n$

converges to $\map f z$ in a neighborhood of $z_0$ in $U$.

Let $U$ be an open subset of $\C$.

Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\C$.

Let $f : U \to X$ be a mapping.

We say that $f$ is analytic if and only if the limit:

$\ds \lim_{w \mathop \to z} \frac {\map f w - \map f z} {w - z}$

exists for each $z \in U$.