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  "body": "\n## Definition\n\nFor real $x$, the *Airy functions* of the first and second kind are\n$$\\operatorname{Ai}(x) = \\frac{1}{\\pi} \\int_0^\\infty \\cos\\left( \\frac{t^3}{3} + x t \\right) dt,$$\n$$\\operatorname{Bi}(x) = \\frac{1}{\\pi} \\int_0^\\infty \\left( \\exp\\left( -\\frac{t^3}{3} + x t \\right) + \\sin\\left( \\frac{t^3}{3} + x t \\right) \\right) dt,$$\nwhere the oscillatory integrals are improper Riemann integrals.\nBoth extend to entire functions of $z \\in \\C$ that solve the [[Airy equation|PNESAD6R5T]] $y'' = z y$.\n\n## Remarks\n\nThe two functions are fixed by their initial values $\\operatorname{Ai}(0) = \\frac{1}{3^{2/3}\\,\\Gamma(2/3)}$, $\\operatorname{Ai}'(0) = -\\frac{1}{3^{1/3}\\,\\Gamma(1/3)}$, $\\operatorname{Bi}(0) = \\frac{1}{3^{1/6}\\,\\Gamma(2/3)}$, $\\operatorname{Bi}'(0) = \\frac{3^{1/6}}{\\Gamma(1/3)}$, where $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\nTheir Wronskian is $\\operatorname{Ai}(z) \\operatorname{Bi}'(z) - \\operatorname{Ai}'(z) \\operatorname{Bi}(z) = \\frac{1}{\\pi}$, so they are independent.\nExample: $\\operatorname{Ai}$ is the solution that tends to $0$ as $x \\to +\\infty$, where $\\operatorname{Ai}(x) \\sim \\frac{e^{-\\zeta}}{2\\sqrt{\\pi}\\, x^{1/4}}$ with $\\zeta = \\frac{2}{3} x^{3/2}$; for $x \u003e 0$ it equals $\\frac{1}{\\pi} \\sqrt{\\frac{x}{3}}\\, K_{1/3}(\\zeta)$ with the [[modified Bessel function|2ET068BZZH]] $K_{1/3}$.\nA non-example: $\\operatorname{Ai} + \\operatorname{Bi}$ solves the equation but is not $\\operatorname{Ai}$, since it grows like $\\operatorname{Bi}$ as $x \\to +\\infty$.\nFor $x \\to -\\infty$, $\\operatorname{Ai}(x) \\sim \\frac{1}{\\sqrt{\\pi}\\, \\abs{x}^{1/4}} \\sin\\left( \\frac{2}{3} \\abs{x}^{3/2} + \\frac{\\pi}{4} \\right)$.\n",
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