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  "body": "\n## Definition\n\nLet $H_n$ be the (physicists') [[Hermite polynomial|MGACY98E9H]] of degree $n \\ge 0$.\nThe *Hermite function* of degree $n$ is the function $\\psi_n \\colon \\R \\to \\R$ given by\n$$\\psi_n(x) = \\left( 2^n \\, n! \\, \\sqrt{\\pi} \\right)^{-1/2} H_n(x) \\, e^{-x^2/2}.$$\n\n## Remarks\n\nExample: $\\psi_0(x) = \\pi^{-1/4} e^{-x^2/2}$, and $\\int_{\\R} \\psi_0(x)^2 \\, dx = \\pi^{-1/2} \\int_{\\R} e^{-x^2} \\, dx = 1$.\nThe orthogonality of the Hermite polynomials for the weight $e^{-x^2}$ gives $\\int_{\\R} \\psi_m \\psi_n \\, dx = \\delta_{mn}$.\nThe family $(\\psi_n)_{n \\ge 0}$ is complete, so it is an orthonormal basis of [[$L^2(\\R)$|SASP14MA1Z]].\nEach $\\psi_n$ solves $-\\psi_n'' + x^2 \\psi_n = (2n + 1) \\psi_n$.\nWith the Fourier transform $\\hat{f}(\\xi) = (2\\pi)^{-1/2} \\int_{\\R} f(x) e^{-i x \\xi} \\, dx$, one has $\\hat{\\psi}_n = (-i)^n \\psi_n$.\nNon-example: the polynomial $H_0 = 1$ without the Gaussian factor is not square integrable on $\\R$, so it is not an element of $L^2(\\R)$ at all.\n",
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