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  "body": "\n## Summary\n\nA rational function $F(s) = P(s)/Q(s)$ with $\\deg P \u003c \\deg Q$ is the [[Laplace transform|0V49JEZW06]] of a sum of terms $t^{k-1} e^{a t}$.\nTo find the [[inverse Laplace transform|07WBTX1MHA]], one splits $F$ by [[partial fraction decomposition|030VFEREKG]] into terms $c/(s - a)^k$ and inverts each term from the rule $\\mathcal{L}\\{t^{k-1} e^{a t}/(k-1)!\\} = 1/(s - a)^k$.\nThis is the everyday method for the transforms that arise from linear differential equations with constant coefficients.\n\n## Explanations\n\n### L1 human\n\nA complicated fraction in $s$ is hard to invert at once, but simple fractions such as $1/(s - a)$ are easy: they come from $e^{a t}$.\nSo one first breaks the fraction into simple pieces and then inverts piece by piece.\nFor example, $\\frac{1}{s (s + 1)} = \\frac{1}{s} - \\frac{1}{s + 1}$, so it is the transform of $1 - e^{-t}$.\nEach root $a$ of the denominator gives a term $e^{a t}$ in the answer, and repeated roots give extra powers of $t$.\n",
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