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  "body": "\n## Definition\n\nLet $\\mathcal{F}^2$ be the space of [[entire functions|0G4F0XB9X2]] $F$ with\n$$\\norm{F}^2 = \\frac{1}{\\pi} \\int_\\C \\abs{F(z)}^2 e^{-\\abs{z}^2}\\, dA(z) \u003c \\infty,$$\nwhere $dA$ is Lebesgue measure on $\\C = \\R^2$.\nIt is a [[Hilbert space|HM2HHE3J51]], the Segal–Bargmann space.\nThe *Bargmann transform* of $f \\in L^2(\\R)$, the [[$L^2$ space|SASP14MA1Z]], is the function\n$$Bf(z) = \\pi^{-1/4} \\int_\\R \\exp\\Big( -\\frac{z^2}{2} + \\sqrt{2}\\, z x - \\frac{x^2}{2} \\Big) f(x)\\, dx, \\qquad z \\in \\C.$$\n\n## Remarks\n\nFor fixed $z$ the kernel is a Gaussian in $x$, so it lies in $L^2(\\R)$, and the integral converges by the Cauchy–Schwarz inequality.\nThe function $Bf$ is entire, and $B$ is a [[unitary operator|MM0EJ2GDE4]] from $L^2(\\R)$ onto $\\mathcal{F}^2$.\nThe monomials $z^n / \\sqrt{n!}$ form an orthonormal basis of $\\mathcal{F}^2$, and $B$ maps the normalized [[Hermite function|52VS07H6GK]] $\\psi_n$ to $z^n / \\sqrt{n!}$.\nConventions differ: some authors use the weight $e^{-\\pi \\abs{z}^2}$ or $e^{-\\abs{z}^2/\\hbar}$ and rescale the kernel accordingly.\n\nExample: $f = \\psi_0 = \\pi^{-1/4} e^{-x^2/2}$.\nThen $Bf(z) = \\pi^{-1/2} e^{-z^2/2} \\int_\\R e^{-x^2 + \\sqrt{2} z x}\\, dx = \\pi^{-1/2} e^{-z^2/2} \\cdot \\sqrt{\\pi}\\, e^{z^2/2} = 1$.\nThe constant $1$ has norm $\\frac{1}{\\pi} \\int_\\C e^{-\\abs{z}^2}\\, dA = 1$, equal to $\\norm{\\psi_0}_{L^2} = 1$.\n\nNon-example: the function $F(z) = e^{z^2}$ is entire but not in $\\mathcal{F}^2$.\nOn the real axis $\\abs{F(x)}^2 e^{-x^2} = e^{x^2}$, and the integral of $\\abs{F}^2 e^{-\\abs{z}^2}$ over a neighbourhood of the real axis diverges, so $F$ is not a Bargmann transform.\n",
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