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  "body": "\n## Statement\n\nLet $\\nu \\ge -\\frac12$, and let $J_\\nu$ be the [[Bessel function of the first kind|CZCMRDWDYW]] of order $\\nu$.\nLet $f \\colon (0, \\infty) \\to \\C$ satisfy $\\int_0^\\infty \\abs{f(r)}\\, r^{1/2}\\, dr \u003c \\infty$, and let $F_\\nu(k) = \\int_0^\\infty f(r) J_\\nu(kr)\\, r\\, dr$ be its [[Hankel transform|2P9YWDGKC4]] of order $\\nu$.\nLet $r \u003e 0$, and assume that $f$ is of bounded variation on a neighbourhood of $r$.\nThen\n$$\\lim_{K \\to \\infty} \\int_0^K F_\\nu(k)\\, J_\\nu(kr)\\, k\\, dk = \\frac{f(r^+) + f(r^-)}{2}.$$\nIn particular, the limit is $f(r)$ at every point $r$ where $f$ is continuous.\n\n## Remarks\n\nThe proof follows the pattern of the Fourier inversion theorem for Fourier integrals.\nAfter exchanging the integrals, the inner integral $\\int_0^K J_\\nu(kr) J_\\nu(kt)\\, k\\, dk$ is computed in closed form from Bessel's equation, and it acts as an approximate identity concentrated at $t = r$ as $K \\to \\infty$.\nA Riemann–Lebesgue argument disposes of the parts away from $r$, and the bounded variation near $r$ controls the part near $r$, exactly as in Dirichlet's theorem for Fourier series.\nThere is also an $L^2$ version: the Hankel transform of order $\\nu$ extends to a unitary operator of $L^2((0, \\infty), r\\, dr)$ that is its own inverse.\n\nExample: for $\\nu = 0$ and $f(r) = e^{-r^2/2}$ one has $F_0(k) = e^{-k^2/2}$, and transforming again gives back $e^{-r^2/2}$.\nFor the indicator function of $[0, 1]$ and $\\nu = 0$, $F_0(k) = J_1(k)/k$, and at $r = 1$ the inversion integral converges to the mean value $\\frac12$ of the jump.\n\n## Explanations\n\n### L1 human\n\nThe Hankel transform breaks a radial function into circular waves.\nThis theorem says that adding the waves back together recovers the function.\nAt a jump, the reconstruction lands exactly halfway between the two values, as for Fourier series.\n\n### L2 human\n\nThe theorem is the radial form of the Fourier inversion theorem, and for $\\nu = \\frac{n}{2} - 1$ it can be derived from it for radial functions on $\\R^n$.\nIt justifies the solution of boundary value problems in cylindrical coordinates: transform in $r$, solve an ordinary differential equation in the remaining variable, and transform back.\nThe hypotheses parallel those of Dirichlet's pointwise convergence theorem for Fourier series.\n",
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