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  "body": "\n## Summary\n\nThe two-sided, or bilateral, Laplace transform of a function $f$ on the whole real line is $\\int_{-\\infty}^{\\infty} f(t) e^{-s t} \\, dt$.\nIt treats past and future times alike, unlike the one-sided [[Laplace transform|0V49JEZW06]], which only sees $t \\ge 0$.\nIt converges in a vertical strip of the complex plane, and on the imaginary axis it becomes the [[Fourier transform|DPVM34ADZV]].\nIn probability it gives the moment generating function, with $s$ replaced by $-s$.\n\n## Explanations\n\n### L1 human\n\nThe ordinary Laplace transform starts the clock at time $0$; the two-sided transform looks at all times, negative and positive.\nBecause of this, it needs $f$ to be small at both ends, and it usually exists only for $s$ in a band between two vertical lines.\nFor example, $e^{-\\abs{t}}$ has two-sided transform $2/(1 - s^2)$ for $-1 \u003c \\operatorname{Re} s \u003c 1$.\n",
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