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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