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  "body": "\n## Definition\n\nLet $E$ be the space of locally integrable functions $f \\colon [0, \\infty) \\to \\C$ of exponential order, that is, with $\\abs{f(t)} \\le M e^{at}$ for some constants $M, a$ and almost all $t$.\nThe *operational calculus* based on the [[Laplace transform|0V49JEZW06]] is the correspondence $f \\mapsto F = \\mathcal{L} f$, $F(s) = \\int_0^\\infty e^{-st} f(t) \\, dt$, used with the following rules.\n\n1. $\\mathcal{L}$ is linear and injective up to equality almost everywhere, so $f$ is recovered from $F$ by the [[inverse Laplace transform|07WBTX1MHA]].\n2. If $f$ is continuous, $f'$ exists except at finitely many points on each bounded interval, and $f' \\in E$, then $\\mathcal{L}(f')(s) = s F(s) - f(0)$.\n3. The [[convolution on the half-line|VDMCEPTTXB]] becomes a product: $\\mathcal{L}(f * g) = \\mathcal{L} f \\cdot \\mathcal{L} g$.\n\nA linear problem for $f$ is solved by transforming it with these rules into an algebraic equation for $F$, solving for $F$, and recovering $f$.\n\n## Remarks\n\nRule 2 is the [[Laplace transform of a derivative|0K090GVHVN]] and rule 3 is the [[convolution theorem|0JNMJEDD93]].\nWith them, a [[linear differential equation|0QQRP17Z7S]] with constant coefficients and given initial values becomes a linear equation for $F$.\nIts coefficients are polynomials in $s$.\nThe name operational calculus also covers other frameworks with the same structure.\nOne is the formal calculus of Heaviside; another is the Mikusiński calculus, where $s$ is an element of a field and no integral transform is used.\n\nExample: $y'' + y = 0$, $y(0) = 0$, $y'(0) = 1$.\nRule 2 twice gives $s^2 Y - 1 + Y = 0$, so $Y(s) = 1/(s^2 + 1)$, which is the transform of $\\sin t$.\nCheck: $\\sin'' + \\sin = 0$, $\\sin 0 = 0$, $\\sin' 0 = 1$.\n\nNon-example: $y' = t\\, e^{t^2}$ fits the scheme only formally.\nIts solutions are $e^{t^2}/2 + C$, which are not of exponential order, so they are outside $E$ and have no Laplace transform.\n",
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