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  "body": "\n## Summary\n\nThe convolution on the half-line combines two functions $f, g$ on $[0, \\infty)$ into $(f * g)(t) = \\int_0^t f(\\tau) g(t - \\tau) \\, d\\tau$.\nIt is the [[convolution on the line|ERP5737VZ1]] of the two functions extended by zero to negative times.\nThe [[Laplace transform|0V49JEZW06]] turns it into a pointwise product.\nIt is also the product of the [[Mikusiński operational calculus|TD5XK5AHNV]].\n\n## Explanations\n\n### L1 human\n\nThink of $t$ as time and of $f$ and $g$ as signals that start at time $0$.\nThe value $(f * g)(t)$ adds up the effects at time $t$ of all the inputs $f(\\tau)$ from earlier times $\\tau$, each weighted by $g$ at the elapsed time $t - \\tau$.\nOnly the past matters, so the integral runs from $0$ to $t$.\nFor example, convolving the constant $1$ with itself gives $\\int_0^t 1 \\, d\\tau = t$.\n",
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