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  "body": "\n## Definition\n\nLet $H$ be the [[Heaviside step function|09C4JP7Y0M]], and consider functions of $t$ that vanish for $t \u003c 0$.\nThe *Heaviside operational calculus* is the formal method that works with the following conventions.\n\n1. The [[differentiation operator|BB15T3J891]] $d/dt$ is written $p$, and $p^{-1}$ means integration from $0$: $(p^{-1} f)(t) = \\int_0^t f(\\tau) \\, d\\tau$.\n2. For each integer $n \\ge 0$, $p^{-n} H = \\frac{t^n}{n!} H$.\n3. Consider a system governed by a [[linear differential equation|0QQRP17Z7S]] with constant coefficients, at rest before $t = 0$. Replace $d/dt$ by $p$ and solve the equation algebraically for $x$. The response $x$ to the input $H$ is then written $x = \\varphi(p) H$ with a rational function $\\varphi$.\n4. $\\varphi(p) H$ is evaluated by expanding $\\varphi(p)$ in powers of $p^{-1}$ and applying rule 2 term by term, or by the expansion theorem.\n\n## Remarks\n\nIn modern terms, $\\varphi(p) H$ is the function whose Laplace transform is $\\varphi(s)/s$, and rule 2 says that $\\mathcal{L}(t^n/n!) = 1/s^{n+1}$.\nFor rational $\\varphi$ that is bounded at infinity, the series in rule 4 converges for all $t$ and the method is correct.\nHeaviside also applied the rules to non-rational expressions such as $\\sqrt{p}$, where they need a separate justification.\n\nExample: $x' - a x = a H$, with $x = 0$ for $t \u003c 0$.\nRule 3 gives $(p - a) x = a H$, so $x = \\frac{a}{p - a} H = \\frac{a/p}{1 - a/p} H = \\sum_{n \\ge 1} a^n p^{-n} H = \\sum_{n \\ge 1} \\frac{a^n t^n}{n!} = e^{at} - 1$ for $t \\ge 0$.\nCheck: $x' - ax = a e^{at} - a e^{at} + a = a$ and $x(0) = 0$.\n\nNon-example: the rules do not apply to an equation with variable coefficients such as $x' - t x = H$.\nThere $p$ does not commute with multiplication by $t$, so the equation cannot be solved for $x$ as $\\varphi(p) H$.\n",
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