Definition:Analytic Function

From ProofWiki
Jump to: navigation, search


Real 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:

$(1): \quad \xi \in \left({c \,.\,.\, d}\right)$
$(2): \quad \displaystyle \forall x \in \left({c \,.\,.\, d}\right): f \left({x}\right) = \sum_{n \mathop = 0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({x}\right)$

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.

Complex Analytic Function

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 $(a_n) : \N \to \C$ such that the series:

$\displaystyle \sum_{n\mathop = 0}^\infty a_n(z-z_0)^n$

converges to $f(z)$ in a neighborhood of $z_0$ in $U$.

Also see