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  "body": "\n## Summary\n\nThe abscissa of convergence of a [[Laplace transform|0V49JEZW06]] is a number $\\sigma_c$ in $[-\\infty, +\\infty]$.\nThe Laplace integral converges on the half-plane $\\operatorname{Re} s \u003e \\sigma_c$ and diverges on the half-plane $\\operatorname{Re} s \u003c \\sigma_c$.\nIt plays the role that the radius of convergence plays for a power series.\nThe same notion exists for [[Dirichlet series|8MWTEF6462]], which are Laplace–Stieltjes transforms of step functions.\nOn the boundary line $\\operatorname{Re} s = \\sigma_c$ the integral may converge at some points and diverge at others.\n\n## Explanations\n\n### L1 human\n\nThe Laplace integral $\\int_0^\\infty f(t) e^{-st}\\, dt$ converges better when $\\operatorname{Re} s$ is larger, because $e^{-st}$ then decays faster.\nSo the set of $s$ where it converges is a right half-plane, possibly the whole plane or empty.\nThe abscissa of convergence is the real part of its left edge.\nFor $f(t) = e^{3t}$ the integral converges exactly when $\\operatorname{Re} s \u003e 3$, so the abscissa is $3$.\n",
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