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  "body": "\n## Summary\n\nThe Y transform of order $\\nu$ is an [[integral transform|YCK709QB06]] on $(0, \\infty)$ with kernel $\\sqrt{xy}\\, Y_\\nu(xy)$.\nHere $Y_\\nu$ is the [[Bessel function of the second kind|GRSNDJG363]].\nIt is the companion of the [[Hankel transform|2P9YWDGKC4]], which uses the Bessel function of the first kind $J_\\nu$ in the same way.\nUnlike the Hankel transform, it is not its own inverse.\nIt appears when boundary value problems in cylindrical coordinates need solutions that are singular on the axis.\n\n## Explanations\n\n### L1 human\n\nBessel functions come in two kinds: $J_\\nu$ is finite at the origin, and $Y_\\nu$ blows up there.\nThe Hankel transform uses $J_\\nu$ as its family of test waves, and the Y transform uses $Y_\\nu$.\nFor the order $\\nu = 1/2$ the kernel is a cosine, so the Y transform is, up to a sign and a constant, the Fourier cosine transform.\nFor other orders it is a genuinely new transform with its own inversion theory.\n",
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