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  "body": "\n## Summary\n\nThe abscissa of absolute convergence $\\sigma_a$ of a [[Laplace transform|0V49JEZW06]] marks the left edge of the half-plane where $\\int_0^\\infty \\abs{f(t)} e^{-\\operatorname{Re}(s) t}\\, dt$ is finite.\nIt is never smaller than the [[abscissa of convergence|FTHFSJ30WJ]] $\\sigma_c$.\nIn the half-plane of absolute convergence the transform can be handled with the standard theorems of Lebesgue integration, such as dominated convergence and Fubini's theorem.\nThe strip $\\sigma_c \u003c \\operatorname{Re} s \u003c \\sigma_a$, where the integral converges only conditionally, can be nonempty.\n\n## Explanations\n\n### L1 human\n\nConvergence of an integral can come from two sources: the size of the integrand gets small, or positive and negative parts cancel.\nAbsolute convergence uses only the size.\nThe abscissa of absolute convergence tells where size alone is enough.\nFor $f(t) = e^{t} \\sin(e^{t})$ the integral converges for $\\operatorname{Re} s \u003e 0$ thanks to cancellation, but converges absolutely only for $\\operatorname{Re} s \u003e 1$.\n",
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