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  "body": "\n## Example\n\nLet $b \u003e 0$ and let $f$ be a given function on $[0, b]$.\n*Abel's integral equation* is the equation\n$$\\int_0^x \\frac{\\varphi(t)}{\\sqrt{x - t}} \\, dt = f(x), \\quad 0 \\le x \\le b,$$\nfor an unknown function $\\varphi$ on $[0, b]$.\nIt is a [[Volterra equation of the first kind|H6ZGEDM4VV]], of [[convolution type|SNFYTC1TTA]], with the [[weakly singular kernel|94TNBAAJYX]] $k(x - t) = (x - t)^{-1/2}$.\nMore generally, Abel's equation with exponent $\\alpha \\in (0, 1)$ has the kernel $(x - t)^{-\\alpha}$.\n\nCheck on an instance: take $\\varphi = 1$.\nThen $\\int_0^x (x - t)^{-1/2} \\, dt = 2 \\sqrt{x}$, so $\\varphi = 1$ solves the equation with $f(x) = 2\\sqrt{x}$.\nTake $\\varphi(t) = t$.\nThe substitution $t = xu$ gives $\\int_0^x t (x - t)^{-1/2} \\, dt = x^{3/2} \\int_0^1 u (1 - u)^{-1/2} \\, du = \\tfrac43 x^{3/2}$, using the [[Beta function|QHNWVSC6AQ]] value $B(2, \\tfrac12) = \\tfrac43$.\n\n## Remarks\n\nThe kernel is not continuous on the diagonal, so the reduction to an equation of the second kind by differentiation does not apply.\nThe solution is instead given by the [[Abel inversion formula|TDMXFY06A2]], $\\varphi(x) = \\frac{1}{\\pi} \\frac{d}{dx} \\int_0^x \\frac{f(t)}{\\sqrt{x - t}} \\, dt$.\nFor $f(x) = 2\\sqrt{x}$ this gives $\\frac{1}{\\pi} \\frac{d}{dx} (\\pi x) = 1$, as expected.\nAbel met this equation in a mechanical problem that generalizes the [[tautochrone problem|A56P5S5D6Z]].\n\n## Explanations\n\n### L1 human\n\nAbel's equation asks to recover a function from a weighted running integral, where the weight $1/\\sqrt{x - t}$ blows up at the current point $x$.\nThe left side is, up to the factor $\\sqrt{\\pi}$, the Riemann–Liouville integral of order one half of $\\varphi$.\nSo solving the equation means taking a \"half derivative\" of $f$.\nIt is one of the oldest integral equations, and it appears in mechanics, in tomography and in the study of fractional calculus.\n",
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