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  "body": "\n## Definition\n\nLet $P$ and $Q$ be complex polynomials with $\\deg P \u003c \\deg Q$, and let $F = P/Q$ be the [[rational function|0WCZH4AFMT]].\nLet $a$ be a [[root|00EHX6FWC0]] of $Q$ of multiplicity $m \\ge 1$, and write $Q(s) = (s - a)^m R(s)$ with $R$ a polynomial and $R(a) \\ne 0$.\nIn the [[partial fraction decomposition|030VFEREKG]] of $F$, let $c$ be the coefficient of $1/(s - a)^m$.\nThe *Heaviside cover-up method* computes this coefficient as\n$$c = \\frac{P(a)}{R(a)} = \\Big[ (s - a)^m F(s) \\Big]_{s = a}.$$\nWhen $a$ is a simple root, this equals $P(a)/Q'(a)$.\n\n## Remarks\n\nThe formula holds because $(s - a)^m F(s)$ equals $c$ plus terms that vanish at $s = a$.\nExample: for $F(s) = \\frac{1}{(s - 1)(s - 2)(s - 3)}$, cover-up gives the coefficients $\\frac{1}{(1-2)(1-3)} = \\tfrac12$, $\\frac{1}{(2-1)(2-3)} = -1$ and $\\frac{1}{(3-1)(3-2)} = \\tfrac12$.\nIts inverse Laplace transform is therefore $\\tfrac12 e^{t} - e^{2t} + \\tfrac12 e^{3t}$.\nNon-example: for a repeated root the method gives only the top coefficient.\nFor $\\frac{1}{s (s - 1)^2} = \\frac{A}{s} + \\frac{B}{s - 1} + \\frac{C}{(s - 1)^2}$, cover-up gives $A = 1$ and $C = 1$.\nThe true value $B = -1$ does not come from covering up; one finds it by comparing coefficients or by differentiating $(s - 1)^2 F(s)$ at $s = 1$.\n",
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    "name": "Heaviside cover-up method",
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  "name": "Heaviside cover-up method",
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