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  "body": "\n## Definition\n\nAn [[entire function|0419794R0T]] $f$ belongs to the *Laguerre–Pólya class* if there is a sequence $(P_k)$ of polynomials such that:\n\n1. each $P_k$ has real coefficients;\n2. all [[roots|00EHX6FWC0]] of each $P_k$ are real;\n3. $P_k \\to f$ [[uniformly|08FPABDQ4C]] on every compact subset of $\\C$.\n\nThe class is written $\\mathcal{L}\\text{-}\\mathcal{P}$.\n\n## Remarks\n\nA function $f$ that is not identically zero belongs to the class if and only if\n$$f(z) = c\\, z^m e^{-a z^2 + b z} \\prod_{n} \\left(1 - \\frac{z}{x_n}\\right) e^{z/x_n}.$$\nHere $c, b \\in \\R$ with $c \\ne 0$, $a \\ge 0$, and $m \\ge 0$ is an integer.\nThe $x_n$ are nonzero real numbers with $\\sum_n x_n^{-2} \u003c \\infty$, and the product is a [[Weierstrass product|ACSF58HYDH]].\nExample: $e^{-z^2}$ is the limit of the polynomials $(1 - z^2/k)^k$, which have real coefficients and only the real roots $\\pm\\sqrt{k}$.\nExample: $\\sin z = z \\prod_{n \\ge 1} (1 - z^2/(n^2\\pi^2))$ is in the class.\nNon-example: $1 + z^2$ has real coefficients but the roots $\\pm i$.\nIt is not a limit of real-rooted real polynomials, because by Hurwitz's theorem such limits, when not identically zero, have only real zeros.\nNon-example: $e^{z^2}$ has no zeros, but its product form would need $a = -1 \u003c 0$.\nThe class is closed under differentiation.\n",
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