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  "body": "\n## Summary\n\nThe Bromwich integral is the complex line integral $\\frac{1}{2 \\pi i} \\int_{\\gamma - i \\infty}^{\\gamma + i \\infty} e^{s t} F(s) \\, ds$ along a vertical line $\\operatorname{Re} s = \\gamma$ in the half-plane where a [[Laplace transform|0V49JEZW06]] $F$ is defined.\nIt is the standard integral formula for the [[inverse Laplace transform|07WBTX1MHA]]: under mild conditions it returns the function $f$ with $\\mathcal{L}\\{f\\} = F$.\nIt also opens the way to computing inverse transforms by [[contour integration|06ZGPYTWK8]] and residues.\n\n## Explanations\n\n### L1 human\n\nThe Laplace transform packs a function of time into a function of a complex variable $s$.\nThe Bromwich integral unpacks it: it adds up the exponentials $e^{s t}$, weighted by $F(s)$, over a vertical line in the complex plane.\nFor example, for $F(s) = 1/s$ and any line $\\operatorname{Re} s = \\gamma \u003e 0$, the integral gives $1$ for $t \u003e 0$ and $0$ for $t \u003c 0$, which is the step function whose transform is $1/s$.\nThe line must lie to the right of all singularities of $F$.\n",
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