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  "body": "\n## Definition\n\nThe *sinc function* is the function $\\operatorname{sinc} \\colon \\R \\to \\R$ given by\n$$\\operatorname{sinc} x = \\begin{cases} \\dfrac{\\sin x}{x} \u0026 \\text{if } x \\ne 0, \\\\ 1 \u0026 \\text{if } x = 0, \\end{cases}$$\nwhere $\\sin$ is the [[sine function|08WFZP6P89]].\nThe *normalised sinc function* is $x \\mapsto \\operatorname{sinc}(\\pi x)$.\n\n## Remarks\n\nThe value at $0$ makes the function continuous, since $\\frac{\\sin x}{x} \\to 1$ as $x \\to 0$.\nIn fact $\\operatorname{sinc} x = \\sum_{k \\ge 0} \\frac{(-1)^k x^{2k}}{(2k+1)!}$, so it extends to an entire even function.\n\nExample: $\\operatorname{sinc}(\\pi/2) = \\frac{1}{\\pi/2} = \\frac{2}{\\pi}$, and the zeros of $\\operatorname{sinc}$ are the points $k\\pi$ with $k$ a non-zero integer.\n\nNon-example: the function equal to $\\frac{\\sin x}{x}$ for $x \\ne 0$ and to $0$ at $x = 0$ is not the sinc function; it fails the second clause and is not continuous at $0$.\n\nThe improper integral $\\int_{-\\infty}^{\\infty} \\operatorname{sinc} x\\, dx = \\pi$ converges, but $\\operatorname{sinc}$ is not Lebesgue integrable on $\\R$.\nSome authors, mainly in signal processing, write $\\operatorname{sinc}$ for the normalised version; Mathlib's `Real.sinc` is the unnormalised one above.\n",
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