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  "body": "\n## Summary\n\nA Liouvillian function is built from rational functions in finitely many steps by taking algebraic functions, integrals and exponentials of integrals.\nThey make precise the idea of a solution \"in closed form\" for differential equations.\n[[Differential Galois theory|EEZS4H8QPH]] decides whether a linear differential equation has Liouvillian solutions.\nSee the [[definition|R80FA7993W]].\n\n## Explanations\n\n### L1 human\n\nStart with fractions of polynomials in $x$.\nAllow three operations again and again: solving polynomial equations, taking antiderivatives, and taking exponentials of antiderivatives.\nEverything one can reach is Liouvillian; for example $e^{x^2}$ and the error function $\\int e^{-x^2} dx$ are.\nElementary functions such as $\\sin x$, $\\ln x$ and $\\sqrt{1 + x^3}$ are all Liouvillian.\nThe solutions of the [[Airy equation|PNESAD6R5T]] $y'' = x y$ are not.\n",
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