# Equivalence of Definitions of Analytic Function

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## Theorem

The following definitions of the concept of **Analytic Function** are equivalent:

### Real Numbers

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 Plane

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 $U$.

## Proof

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