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  "body": "\n## Example\n\nLet $a \\ge 0$ and let $\\delta_a$ be the [[Dirac delta|0KPHENZ5S7]] at $a$, that is, the [[Dirac measure|099SXAG014]] that gives mass $1$ to the point $a$.\nThe delta is not a function, so its Laplace transform is defined as the [[Laplace–Stieltjes transform|KHRTBP37VD]] of the measure.\nThe point $0$ is included in the domain of integration:\n$$\\mathcal{L}\\{\\delta_a\\}(s) = \\int_{[0, \\infty)} e^{-st}\\, d\\delta_a(t) = e^{-as} \\quad (s \\in \\C).$$\nCheck: integrating a continuous function against $\\delta_a$ evaluates it at $a$, and $t \\mapsto e^{-st}$ is continuous.\nIn particular $\\mathcal{L}\\{\\delta_0\\} = 1$, with the convention that the lower limit is $0^-$.\nThis is consistent with the [[Laplace transform|05XT5AVJBK]] of the step $H(t - a)$, which is $e^{-as}/s$.\nIndeed, the delta is the derivative of the step, and differentiation multiplies the transform by $s$ when the initial value is $0$.\n\n## Explanations\n\n### L1 human\n\nThe delta models an instant kick of total size $1$ at time $a$.\nIts transform is $e^{-as}$, and for a kick at time $0$ it is just the constant $1$.\nSince the transform of an ordinary function tends to $0$ as $s \\to \\infty$, the constant $1$ also shows that the delta is not an ordinary function.\nIn system theory, the response to the kick $\\delta_0$ is the impulse response, and its transform is the transfer function.\n",
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