Analytic

Real Analyticity

A function $f(x)$ is analytic at a point $a$ in its domain if it satisfies the two following conditions:

1. On some open interval around $a$ in the real domain, $(b,c)\in \mathbb{R}$, $f(x)$ can be expressed as its convergent Taylor series,¹

$$f(x)=\sum _{n=0}^{\infty }{c}_n(x-a{)}^n=\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n\text{\hspace{1em}(1)}$$

2. $f(x)$ is infinitely differentiable everywhere on its domain, meaning that it has derivatives of every order which exist at all points on its domain.

Footnotes

1. See The Taylor Series. ↩