Analytic function

Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$. The function $f$ is analytic at $a \in I$ if there are $r > 0$ and a power series $\sum c_n (x - a)^n$ that converges to $f(x)$ for all $x$ with $|x - a| < r$. It is analytic on $I$ if it is analytic at each point of $I$.

Remarks

An analytic function is smooth, and its power series at $a$ is its Taylor series at $a$. Polynomials, the exponential function, sine, and cosine are analytic on $\mathbb{R}$. A smooth function need not be analytic. See the smooth non-analytic function. Mathlib defines analyticity at a point as AnalyticAt.

Explanations

L1 human

An analytic function equals a convergent power series near each point. So it is determined near a point by its derivatives at that point.