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  "body": "\n## Definition\n\nLet $I$ be an open interval and let $f \\colon I \\to \\mathbb{R}$.\nThe function $f$ is analytic at $a \\in I$ if there are $r \u003e 0$ and a [[power series|0VP56B4EMX]] $\\sum c_n (x - a)^n$ that converges to $f(x)$ for all $x$ with $|x - a| \u003c r$.\nIt is analytic on $I$ if it is analytic at each point of $I$.\n\n## Remarks\n\nAn analytic function is [[smooth|0BE3QDH73A]], and its power series at $a$ is its [[Taylor series|0225QPWRRE]] at $a$.\nPolynomials, the exponential function, sine, and cosine are analytic on $\\mathbb{R}$.\nA smooth function need not be analytic. See the [[smooth non-analytic function|09RTYYHXKA]].\nMathlib defines analyticity at a point as `AnalyticAt`.\n\n## Explanations\n\n### L1 human\n\nAn analytic function equals a convergent power series near each point.\nSo it is determined near a point by its derivatives at that point.\n",
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