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  "body": "\n## Description\n\nLet $C_c^\\infty(\\mathbb{R})$ be the space of infinitely differentiable functions $\\varphi : \\mathbb{R} \\to \\mathbb{C}$ that vanish outside a bounded set.\nThe Dirac delta is the distribution $\\delta$ that maps each test function $\\varphi \\in C_c^\\infty(\\mathbb{R})$ to $\\delta[\\varphi] = \\varphi(0)$.\nInformally, one writes $\\int_{-\\infty}^{\\infty} \\delta(t)\\, \\varphi(t)\\, dt = \\varphi(0)$, as if $\\delta$ were a function that is $0$ away from $0$ and has total integral $1$.\nNo actual function has this property. The delta is a generalized function.\n\n## Remarks\n\nAs a measure, $\\delta$ gives the mass $1$ to each set that contains $0$ and the mass $0$ to the other sets.\nThe shifted delta $\\delta(t - a)$ maps $\\varphi$ to $\\varphi(a)$. For $a \u003e 0$, its Laplace transform is $e^{-a s}$, and for $a = 0$ the usual convention gives $1$.\nIn the sense of distributions, $\\delta$ is the derivative of the [[Heaviside step function|09C4JP7Y0M]].\nIn differential equations, $\\delta(t - a)$ models an impulse: a large force that acts for a very short time around $t = a$.\n\n## Explanations\n\n### L1 human\n\nThe delta function is an idealized spike at $0$, infinitely tall and infinitely thin, with area $1$.\nIt is not a real function. But it makes sense when we integrate it against a smooth function, and the result is the value of that function at $0$.\n",
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