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  "body": "\n## Definition\n\nLet $F$ be a [[meromorphic function|0NSD1TPHR0]] on $\\C$ with finitely many poles $s_1, \\ldots, s_n$.\nLet $\\gamma$ be a real number with $\\gamma \u003e \\operatorname{Re} s_k$ for every $k$.\nAssume that there are constants $C \u003e 0$, $p \u003e 0$ and $R_0 \u003e 0$ with $\\abs{F(s)} \\le C \\abs{s}^{-p}$ for all $s$ with $\\abs{s} \\ge R_0$ and $\\operatorname{Re} s \\le \\gamma$.\n*Residue calculus for Laplace inversion* is the evaluation, for $t \u003e 0$, of the [[Bromwich integral|9WRE8KEJJX]] of $F$ along $\\operatorname{Re} s = \\gamma$ as\n$$\\frac{1}{2 \\pi i} \\int_{\\gamma - i \\infty}^{\\gamma + i \\infty} e^{s t} F(s) \\, ds = \\sum_{k=1}^{n} \\operatorname{Res}_{s = s_k} \\left( e^{s t} F(s) \\right),$$\nwhere $\\operatorname{Res}$ is the [[residue|0JFGKKWWFK]].\nUnder these assumptions the identity holds for every $t \u003e 0$.\nLet $F$ also be the [[Laplace transform|05XT5AVJBK]] of a function $f$ as in Mellin's inverse formula.\nThen the right side is the [[inverse Laplace transform|0JRW4X1MDC]] $f(t)$ at each $t \u003e 0$ where $f$ is continuous.\n\n## Remarks\n\nThe identity follows from the [[residue theorem|032HKYS329]] on the region bounded by the segment from $\\gamma - i R$ to $\\gamma + i R$ and a left arc of radius about $R$.\nA Jordan-type estimate shows that the arc contributes $0$ in the limit when $t \u003e 0$.\nThe hypotheses hold for every rational $F = P/Q$ with $\\deg P \u003c \\deg Q$.\nExample: $F(s) = 1/(s + 1)^2$ has one double pole, and $\\operatorname{Res}_{s=-1} e^{s t}/(s+1)^2 = \\frac{d}{ds} e^{s t} \\big|_{s = -1} = t e^{-t}$.\nNon-example: $F(s) = e^{-s}/s$ is not small on left arcs, because $\\abs{e^{-s}} = e^{-\\operatorname{Re} s}$ grows there.\nThe residue sum would give $1$ for all $t \u003e 0$, while the true inverse is the delayed step $u(t - 1)$, which is $0$ for $0 \u003c t \u003c 1$.\nTransforms that come from partial differential equations often have infinitely many poles.\nFor them one uses a sequence of contours that avoid the poles, and one gets a series over all poles.\n",
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