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  "body": "\n## Summary\n\nThe Hankel transform of order $\\nu$ of a function $f$ on $(0, \\infty)$ is $\\int_0^\\infty f(r)\\, J_\\nu(kr)\\, r\\, dr$, where $J_\\nu$ is the [[Bessel function of the first kind|CZCMRDWDYW]].\nIt is the radial part of the [[Fourier transform|DPVM34ADZV]].\nThe Fourier transform of a radial function in the plane is a Hankel transform of order $0$, and in higher dimensions other orders appear.\nWith this normalization it is its own inverse.\nIt is used to solve problems with circular or cylindrical symmetry, such as heat flow and vibrations of a drum.\n\n## Explanations\n\n### L1 human\n\nThe Fourier transform writes a function as a sum of plane waves.\nFor a function that only depends on the distance to a centre, circular waves are better suited, and their profiles are Bessel functions.\nThe Hankel transform measures how much of each circular wave of frequency $k$ the function contains.\nFor example, the Gaussian $e^{-r^2/2}$ has the Hankel transform $e^{-k^2/2}$ of order $0$, the same shape again.\n",
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