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  "body": "\n## Summary\n\nParabolic cylinder functions are solutions of Weber's equation $y'' + \\left( \\nu + \\frac12 - \\frac{z^2}{4} \\right) y = 0$.\nThe standard solution $D_\\nu(z)$ decays like $z^\\nu e^{-z^2/4}$ as $z \\to +\\infty$.\nThey arise when the [[Helmholtz equation|MNXKTXJ35G]] or [[Laplace's equation|S8RG6N1PRT]] is solved by [[separation of variables|47NFRVB564]] in parabolic cylindrical coordinates.\nFor integer $\\nu \\ge 0$ they reduce to [[Hermite functions|52VS07H6GK]].\nThey are [[confluent hypergeometric functions|6AZ4ZBN1M8]] in the variable $z^2/2$.\n\n## Explanations\n\n### L1 human\n\nWeber's equation is the Schrödinger equation of the quantum harmonic oscillator, with energy parameter $\\nu + \\frac12$.\nFor most values of $\\nu$ every solution blows up at $+\\infty$ or at $-\\infty$.\nThe function $D_\\nu$ is the solution that decays at $+\\infty$.\nFor $\\nu = 0$ it is the Gaussian $e^{-z^2/4}$: then $y'' = \\left( \\frac{z^2}{4} - \\frac12 \\right) y$, which is exactly the equation with $\\nu = 0$.\nFor $\\nu = n$ a nonnegative integer, $D_n$ decays at both ends, and these are the bound states of the oscillator.\n",
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