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  "body": "\n## Definition\n\nFix a real number $\\nu$, and let $Y_\\nu$ be the [[Bessel function of the second kind|GRSNDJG363]] of order $\\nu$.\nLet $f \\colon (0, \\infty) \\to \\C$ be such that the integral below converges for $y$ in a set $Y \\subseteq (0, \\infty)$.\nThe *Y transform* of order $\\nu$ of $f$ is the function\n$$(\\mathcal{Y}_\\nu f)(y) = \\int_0^\\infty f(x)\\, Y_\\nu(xy)\\, \\sqrt{xy}\\, dx, \\qquad y \\in Y.$$\n\n## Remarks\n\nExample: $Y_{1/2}(z) = -\\sqrt{2/(\\pi z)}\\, \\cos z$, because $Y_{1/2} = -J_{-1/2}$ and $J_{-1/2}(z) = \\sqrt{2/(\\pi z)}\\, \\cos z$.\nSo $\\sqrt{xy}\\, Y_{1/2}(xy) = -\\sqrt{2/\\pi}\\, \\cos(xy)$, and $\\mathcal{Y}_{1/2} f(y) = -\\sqrt{2/\\pi} \\int_0^\\infty f(x) \\cos(xy)\\, dx$, the negative of the Fourier cosine transform.\n\nNon-example: for $\\nu \u003e 0$, $Y_\\nu(z)$ behaves like $-\\frac{\\Gamma(\\nu)}{\\pi} (2/z)^{\\nu}$ as $z \\to 0^+$.\nFor $\\nu = 2$ and $f$ equal to $1$ on $(0, 1]$ and $0$ elsewhere, the integrand behaves like a constant times $x^{-3/2}$ near $0$, so the integral diverges for every $y \u003e 0$.\n\nThe [[Hankel transform|2P9YWDGKC4]] is defined in the same way with $J_\\nu$ in place of $Y_\\nu$.\n",
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