# Category:Analytic Functions

This category contains results about **Analytic Functions**.

Definitions specific to this category can be found in Definitions/Analytic Functions.

### Real Analytic Function

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 described as being **analytic** at the point $\xi$.

That is, a function is **analytic** at a point if and only 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 $\sequence {a_n}: \N \to \C$ 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 $U$.

### Banach Space Valued Function

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 $z \in U$.

## Subcategories

This category has only the following subcategory.

### A

- Analytic Complex Functions (empty)

## Pages in category "Analytic Functions"

The following 3 pages are in this category, out of 3 total.