Definition:Analytic Function/Real Numbers

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 $\xi$.

That is, a real function is an analytic real function at a point $\xi$ if and only if it equals its Taylor series expansion in some open interval containing $\xi$.

Space of Analytic Real Functions

The set of all analytic real functions is called the space of analytic (real) functions and is denoted $\mathrm C^\omega$.

Also known as

This is also known as a real analytic function.

Also see

• Results about analytic real functions can be found here.

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 15.4$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): analytic: 2. (of a real function)
• 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.3.2$: Power series: $(1.42)$