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  "body": "\n## Statement\n\nLet $f \\colon [0, \\infty) \\to \\C$ be continuous with $\\int_0^\\infty \\abs{f(r)}\\, dr \u003c \\infty$.\nLet $g(x, y) = f\\big(\\sqrt{x^2 + y^2}\\big)$ be the radial function on $\\R^2$ with profile $f$.\nThen for every $y \\in \\R$ the integral $\\int_\\R g(x, y)\\, dx$ converges absolutely, and\n$$\\int_\\R g(x, y)\\, dx = F(\\abs{y}),$$\nwhere $F$ is the [[Abel transform|QMW1MWQH6J]] of $f$.\n\n## Remarks\n\nSince $g(x, y)$ is even in $x$, the line integral is $2 \\int_0^\\infty f\\big(\\sqrt{x^2 + y^2}\\big)\\, dx$.\nFor $y \\ne 0$, the substitution $r = \\sqrt{x^2 + y^2}$, with $dx = \\frac{r\\, dr}{\\sqrt{r^2 - y^2}}$, maps $x \\in (0, \\infty)$ onto $r \\in (\\abs{y}, \\infty)$ and gives the formula; for $y = 0$ it is immediate.\nThe absolute convergence follows the same way: $\\frac{r}{\\sqrt{r^2 - y^2}}$ is integrable near $r = \\abs{y}$ and tends to $1$ as $r \\to \\infty$, while $f$ is bounded near $\\abs{y}$ and integrable.\nBy rotation invariance, the integral of $g$ along any line at distance $\\abs{y}$ from the origin is also $F(\\abs{y})$; this is the [[Radon transform|VJNZJ352PC]] of $g$.\n\nExample: for $f(r) = e^{-r^2}$ the line integral is $\\int_\\R e^{-x^2 - y^2}\\, dx = \\sqrt{\\pi}\\, e^{-y^2}$, which is the Abel transform of $f$ at $\\abs{y}$.\n\n## Explanations\n\n### L1 human\n\nA round object seen from the side looks the same from every direction.\nThe Abel transform gives what one sees: the total along each line of sight as a function of the distance of the line from the centre.\nThis statement is the computation behind that picture.\n\n### L2 human\n\nThe statement identifies the Abel transform with the two-dimensional Radon transform restricted to radial functions.\nTogether with the projection-slice theorem, it shows that the Fourier transform of the Abel transform of $f$ is the [[Hankel transform|2P9YWDGKC4]] of order $0$ of $f$, up to constants.\nThis chain is used in tomography of axially symmetric objects, where only one projection is needed.\n",
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