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  "body": "\n## Definition\n\nLet $\\nu$ be a complex number.\n*Weber's equation* with parameter $\\nu$ is the [[linear differential equation|0QQRP17Z7S]]\n$$\\frac{d^2 y}{dz^2} + \\left( \\nu + \\frac12 - \\frac{z^2}{4} \\right) y = 0.$$\nLet $M(a, b, x)$ be [[Kummer's function|6AZ4ZBN1M8]] and $\\Gamma$ the [[gamma function|0H5SYBSED9]], with $1/\\Gamma$ read as $0$ at the poles of $\\Gamma$.\nThe *parabolic cylinder function* $D_\\nu$ is the entire function\n$$D_\\nu(z) = 2^{\\nu/2} e^{-z^2/4} \\left( \\frac{\\sqrt{\\pi}}{\\Gamma\\left( \\frac{1 - \\nu}{2} \\right)} M\\left( -\\frac{\\nu}{2}, \\frac12, \\frac{z^2}{2} \\right) - \\frac{\\sqrt{2\\pi}\\, z}{\\Gamma\\left( -\\frac{\\nu}{2} \\right)} M\\left( \\frac{1 - \\nu}{2}, \\frac32, \\frac{z^2}{2} \\right) \\right).$$\n\n## Remarks\n\nThe function $D_\\nu$ solves Weber's equation, and it is the solution with $D_\\nu(z) \\sim z^\\nu e^{-z^2/4}$ as $z \\to +\\infty$.\nThe functions $D_\\nu(z)$ and $D_\\nu(-z)$ are solutions too, and they are linearly independent unless $\\nu$ is a nonnegative integer.\nAnother common notation is $U(a, z) = D_{-a - \\frac12}(z)$.\nExample: for $\\nu = 0$, the first coefficient is $\\sqrt{\\pi}/\\Gamma(\\frac12) = 1$ and the second is $0$, since $1/\\Gamma(0) = 0$; so $D_0(z) = e^{-z^2/4}$.\nCheck: $y = e^{-z^2/4}$ gives $y'' = \\left( \\frac{z^2}{4} - \\frac12 \\right) y$, so $y'' + \\left( \\frac12 - \\frac{z^2}{4} \\right) y = 0$.\nMore generally, for $\\nu = n \\in \\N$, one has $D_n(z) = 2^{-n/2} e^{-z^2/4} H_n(z / \\sqrt{2})$ with $H_n$ the [[Hermite polynomials|MGACY98E9H]].\nNon-example: $y = e^{z^2/4}$ solves Weber's equation for $\\nu = -1$, since $y'' = \\left( \\frac12 + \\frac{z^2}{4} \\right) y$, but it is not $D_{-1}$: it grows as $z \\to +\\infty$ instead of decaying.\n",
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