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  "body": "\n## Summary\n\nThe [[Laplace transform|0V49JEZW06]] extends from functions to distributions supported in $[0, \\infty)$ that do not grow faster than an exponential.\nThe extension gives meaning to transforms of generalized functions such as the [[Dirac delta|0KPHENZ5S7]] and its derivatives: $\\mathcal{L}\\delta = 1$ and $\\mathcal{L}\\delta^{(k)} = s^k$.\nIt makes the rule \"differentiation becomes multiplication by $s$\" exact, with no initial-value terms, which is the natural setting for impulses in operational calculus.\n\n## Explanations\n\n### L1 human\n\nEngineers like to feed a system an instantaneous kick, the Dirac delta, which is not a genuine function.\nThe ordinary Laplace transform cannot handle it, but the distributional version can, and it gives the delta the transform $1$.\nThe jump of a switched-on signal is then accounted for by a delta in its derivative.\nFor example, the unit step has transform $1/s$, and its derivative, the delta, has transform $s \\cdot 1/s = 1$.\n",
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