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  "body": "\n## Description\n\nThe Heaviside step function is the function $H : \\mathbb{R} \\to \\mathbb{R}$ with $H(x) = 0$ for $x \u003c 0$ and $H(x) = 1$ for $x \u003e 0$.\nConventions for the value $H(0)$ differ: common choices are $0$, $\\frac{1}{2}$ and $1$.\nFor $a \\ge 0$ and $s \\in \\mathbb{C}$ with $\\operatorname{Re} s \u003e 0$, the [[Laplace transform|05XT5AVJBK]] of $t \\mapsto H(t - a)$ is $e^{-a s} / s$.\n\n## Remarks\n\nThe function $t \\mapsto H(t - a)$ switches a signal on at the time $a$. For example, $f(t) H(t - a)$ is $0$ before $a$ and $f(t)$ after $a$.\nFor $g : [0, \\infty) \\to \\mathbb{C}$ with a Laplace transform $G$ and $a \\ge 0$, the shifted function $t \\mapsto H(t - a)\\, g(t - a)$ has the transform $e^{-a s} G(s)$.\nIn the sense of distributions, the derivative of $H$ is the [[Dirac delta|0KPHENZ5S7]].\nIt is named after Oliver Heaviside.\n\n## Explanations\n\n### L1 human\n\nThe step function is off before time $0$ and on after it.\nIt models a switch, such as a voltage turned on at one instant.\n",
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