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  "body": "\n## Statement\n\nLet $P$ and $Q$ be polynomials with complex coefficients with $\\deg P \u003c \\deg Q$.\nSuppose that $Q$ has $n = \\deg Q$ distinct roots $a_1, \\dots, a_n$.\nThen $P/Q$ is the [[Laplace transform|0V49JEZW06]], for $\\operatorname{Re} s \u003e \\max_k \\operatorname{Re} a_k$, of the function\n$$f(t) = \\sum_{k=1}^n \\frac{P(a_k)}{Q'(a_k)} \\, e^{a_k t}, \\qquad t \\ge 0.$$\n\n## Remarks\n\nSince the roots are simple, the [[partial fraction decomposition|030VFEREKG]] of $P/Q$ is $\\sum_k c_k / (s - a_k)$.\nMultiplying by $s - a_k$ and letting $s \\to a_k$ gives $c_k = P(a_k) / Q'(a_k)$, because $Q(s)/(s - a_k) \\to Q'(a_k)$.\nEach term $c_k/(s - a_k)$ is the Laplace transform of $c_k e^{a_k t}$ on $\\operatorname{Re} s \u003e \\operatorname{Re} a_k$.\n\nExample: $P = 1$, $Q(s) = s^2 + 1$, with roots $\\pm i$ and $Q'(s) = 2s$.\nThe theorem gives $f(t) = \\frac{e^{it}}{2i} + \\frac{e^{-it}}{-2i} = \\sin t$, the [[inverse Laplace transform|07WBTX1MHA]] of $1/(s^2 + 1)$.\n\nIn Heaviside's form, the response to a unit step is $\\varphi(p) H$ with $\\varphi = P/Q$, $\\deg P \\le \\deg Q$ and $Q(0) \\ne 0$.\nIf the roots $a_k$ of $Q$ are simple, the theorem applied to $P(s)/(s Q(s))$ gives, for $t \\ge 0$,\n$$\\varphi(p) H = \\frac{P(0)}{Q(0)} + \\sum_k \\frac{P(a_k)}{a_k Q'(a_k)} e^{a_k t}.$$\n\nThe hypothesis of simple roots is needed: $1/s^2$ is the transform of $t$, which is not a combination of exponentials.\nFor repeated roots, the residue formula gives terms $t^j e^{a_k t}$ instead.\n\n## Explanations\n\n### L1 human\n\nA rational function of $s$ breaks into simple pieces $c/(s - a)$, and each piece is the transform of an exponential $c\\,e^{at}$.\nThe theorem gives a direct recipe for the numbers $c$, without solving a system of equations.\nSo the inverse transform of a rational function with simple poles can be written down at once.\n\n### L2 human\n\nThe coefficient $P(a_k)/Q'(a_k)$ is the residue of $P/Q$ at $a_k$, and the theorem is the special case of the residue form of the inverse Laplace transform, $f(t) = \\sum_k \\operatorname{Res}_{s = a_k} \\left( e^{st} P(s)/Q(s) \\right)$.\nIt is the main tool for solving linear constant-coefficient systems by the [[Heaviside operational calculus|QQ98C2AXZP]] or by the Laplace transform, because their transfer functions are rational.\nThe exponents $a_k$ are the natural frequencies of the system, and the theorem expresses the response as a superposition of its modes.\n",
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