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  "body": "\n## Definition\n\nLet $f(z) = \\sum_{k \\ge 0} c_k z^k$ be a [[formal power series|072W67KFYK]] with complex coefficients, and let $m, n \\ge 0$ be integers.\nConsider pairs $(P, Q)$ of polynomials with complex coefficients such that:\n\n1. $P$ has [[degree|0DPF6WRZW6]] at most $m$;\n2. $Q$ is not the zero polynomial and has degree at most $n$;\n3. the series $Q(z) f(z) - P(z)$ has no terms of degree at most $m + n$, that is, $Q f - P = O(z^{m+n+1})$.\n\nSuch pairs exist, and the [[rational function|0WCZH4AFMT]] $P/Q$ is the same for all of them.\nThis rational function is the *$[m/n]$ Padé approximant* of $f$, written $[m/n]_f$.\n\n## Remarks\n\nExistence: condition 3 for the coefficients of $z^{m+1}, \\dots, z^{m+n}$ is a system of $n$ homogeneous linear equations in the $n + 1$ coefficients of $Q$.\nSo a nonzero $Q$ exists, and then $P$ is the truncation of $Q f$ at degree $m$.\nUniqueness: for two pairs, $P Q' - P' Q = Q'(Q f - P) - Q(Q' f - P')$ is a polynomial of degree at most $m + n$ that is $O(z^{m+n+1})$, so it is $0$.\nExample: for $f = e^z$, $m = n = 1$, take $Q = 1 - z/2$ and $P = 1 + z/2$.\nThen $Q f = 1 + z/2 + 0 \\cdot z^2 - z^3/12 + \\cdots$, so $Q f - P = O(z^3)$ and $[1/1]_{e^z} = (2 + z)/(2 - z)$.\nNon-example: $P = 1 + z$ and $Q = 1$ satisfy clauses 1 and 2 for $m = n = 1$, but $Q f - P = z^2/2 + \\cdots$ fails clause 3.\nFor $n = 0$ the approximant $[m/0]_f$ is the [[Taylor polynomial|0HFSFVMWQB]] of degree $m$.\nWhen $Q(0) \\ne 0$ one usually normalizes $Q(0) = 1$, and then $f - P/Q = O(z^{m+n+1})$; when $Q(0) = 0$ the approximant may match fewer coefficients.\n",
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