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  "body": "\n## Definition\n\nFor real $x$, *Scorer's functions* are\n$$\\operatorname{Gi}(x) = \\frac{1}{\\pi} \\int_0^\\infty \\sin\\left( \\frac{t^3}{3} + x t \\right) dt, \\qquad \\operatorname{Hi}(x) = \\frac{1}{\\pi} \\int_0^\\infty \\exp\\left( -\\frac{t^3}{3} + x t \\right) dt,$$\nwhere the first integral is an improper Riemann integral.\n\n## Remarks\n\nThey solve the inhomogeneous [[Airy equations|PNESAD6R5T]] $\\operatorname{Gi}'' - x \\operatorname{Gi} = -\\frac{1}{\\pi}$ and $\\operatorname{Hi}'' - x \\operatorname{Hi} = \\frac{1}{\\pi}$.\nCheck for $\\operatorname{Hi}$: differentiating under the integral sign, $\\operatorname{Hi}'' - x \\operatorname{Hi} = \\frac{1}{\\pi} \\int_0^\\infty (t^2 - x)\\, e^{-t^3/3 + xt}\\, dt = -\\frac{1}{\\pi} \\left[ e^{-t^3/3 + xt} \\right]_0^\\infty = \\frac{1}{\\pi}$.\nBy comparing with the integral that defines the [[Airy function|1N5D4PN92W]] $\\operatorname{Bi}$, one has $\\operatorname{Gi} + \\operatorname{Hi} = \\operatorname{Bi}$.\nA non-example: $\\operatorname{Bi}$ itself is not a Scorer function; it solves the homogeneous equation $y'' - x y = 0$, not one with right-hand side $\\pm \\frac{1}{\\pi}$.\nBoth functions extend to entire functions of $z \\in \\C$.\n",
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