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  "body": "\n## Definition\n\nLet $K$ be a [[differential field|T43TQ817XC]] of characteristic $0$ with derivation $a \\mapsto a'$.\nLet $L \\supseteq K$ be a differential field extension with the same field of constants as $K$.\nIt is *Liouvillian* if there is a tower of differential fields $K = K_0 \\subseteq \\dots \\subseteq K_m = L$ with the following property.\nFor each $i$, $K_i = K_{i-1}(t_i)$, where $t_i$ satisfies one of these conditions:\n\n1. $t_i$ is algebraic over $K_{i-1}$;\n2. $t_i' \\in K_{i-1}$, so $t_i$ is an integral of an element of $K_{i-1}$;\n3. $t_i \\ne 0$ and $t_i'/t_i \\in K_{i-1}$, so $t_i$ is an exponential of an integral of an element of $K_{i-1}$.\n\nLet $\\C(x)$ be the field of rational functions with the derivation $d/dx$.\nA *Liouvillian function* is an element of a Liouvillian extension of $\\C(x)$, realized as a field of meromorphic functions on some domain.\n\n## Remarks\n\nExample: $t = e^{x^2}$ is Liouvillian over $\\C(x)$, since $t'/t = 2x \\in \\C(x)$ (clause 3).\nAn antiderivative of $e^{-x^2}$ is Liouvillian too: adjoin $e^{-x^2}$ by clause 3 and then the integral by clause 2.\nAlso $\\ln x$ is Liouvillian by clause 2, since $(\\ln x)' = 1/x$, and $\\sin x = (e^{ix} - e^{-ix})/(2i)$ is Liouvillian.\nNon-example: no nonzero solution of the Airy equation $y'' = x y$ is Liouvillian; this is a result of differential Galois theory, since the differential Galois group of that equation is $\\mathrm{SL}_2(\\C)$.\nElementary functions are the special case in which clause 2 is restricted to logarithms and clause 3 to exponentials.\nSo every elementary function is Liouvillian, but $\\int e^{-x^2} dx$ is Liouvillian and not elementary, by [[Liouville's theorem|8VRTPECBPZ]].\n",
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