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  "body": "\n## Definition\n\nLet $p$ be a prime number.\nLet $\\Q_p$ be the field of [[$p$-adic numbers|BT6T702C28]], the completion of $\\Q$ for the [[$p$-adic absolute value|WGN82TCEF2]] $\\abs{x}_p = p^{-v_p(x)}$, where $v_p$ is the [[$p$-adic valuation|03HDF50VN4]].\n*$p$-adic analysis* is the study of functions defined on subsets of $\\Q_p$ or of a complete field extension of $\\Q_p$.\nThe values lie in such a field or in $\\R$ or $\\C$.\nIt treats limits, series, continuity, differentiability, power series and integration for the topology given by $\\abs{\\cdot}_p$.\n\n## Remarks\n\nThe absolute value $\\abs{\\cdot}_p$ satisfies the strong triangle inequality $\\abs{x + y}_p \\le \\max(\\abs{x}_p, \\abs{y}_p)$.\nHence a series $\\sum_n a_n$ in $\\Q_p$ converges if and only if $a_n \\to 0$.\n\nExample: $\\abs{p^n}_p = p^{-n} \\to 0$, so $\\sum_{n \\ge 0} p^n$ converges in $\\Q_p$, and its sum is $\\frac{1}{1 - p}$, since $(1 - p) \\sum_{n=0}^{N} p^n = 1 - p^{N+1} \\to 1$.\n\nNon-example: the same series $\\sum_{n \\ge 0} p^n$ diverges for the usual absolute value on $\\R$; its study is part of real analysis, not of $p$-adic analysis.\n\nTwo typical facts.\nThe ring $\\Z_p$ of [[$p$-adic integers|V6TJAFZ4MD]] is compact and totally disconnected.\nThe exponential series $\\sum_n x^n / n!$ converges in $\\Q_p$ exactly when $\\abs{x}_p \u003c p^{-1/(p-1)}$.\n",
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