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  "body": "\n## Summary\n\nA function $f$ on $[0, \\infty)$ is of exponential order when it grows no faster than some exponential $M e^{at}$ for large $t$.\nThis is the usual sufficient condition for the [[Laplace transform|0V49JEZW06]] of $f$ to exist.\nFor a piecewise continuous $f$ of exponential order $a$, the integral $\\int_0^\\infty f(t) e^{-st}\\, dt$ converges absolutely for $\\operatorname{Re} s \u003e a$.\nMost functions met in linear differential equations, such as polynomials, exponentials, sines and cosines, are of exponential order.\n\n## Explanations\n\n### L1 human\n\nThe factor $e^{-st}$ in the Laplace transform decays exponentially.\nIt can tame $f$ only if $f$ itself grows at most exponentially.\nFor example, $t^3$ is of exponential order, because $t^3 \\le 6 e^t$ for all $t \\ge 0$.\nThe function $e^{t^2}$ is not: it eventually beats every $e^{at}$, and indeed it has no Laplace transform.\n",
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