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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers, possibly $b = \\infty$, and let $f$ be a locally integrable real function on $[a, b)$.\nLet $\\alpha \u003e 0$ be real.\nThe *Riemann–Liouville integral* of order $\\alpha$ of $f$, with lower terminal $a$, is the function\n$$I_a^\\alpha f(x) = \\frac{1}{\\Gamma(\\alpha)} \\int_a^x (x - t)^{\\alpha - 1} f(t) \\, dt, \\qquad a \u003c x \u003c b,$$\nwhere $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\n\n## Remarks\n\nThe integral converges for almost every $x$, and for every $x$ when $f$ is continuous on $[a, b)$, because $(x - t)^{\\alpha - 1}$ is integrable near $t = x$ for $\\alpha \u003e 0$.\nExample: for $\\beta \u003e -1$ and $a = 0$, the substitution $t = xs$ and the beta integral give $I_0^\\alpha t^\\beta = \\frac{\\Gamma(\\beta + 1)}{\\Gamma(\\alpha + \\beta + 1)} x^{\\alpha + \\beta}$.\nFor $\\beta = 0$ this is $x^\\alpha / \\Gamma(\\alpha + 1)$.\nNon-example: for $\\alpha = -\\tfrac{1}{2}$ and $f = 1$, the integrand $(x - t)^{-3/2}$ is not integrable near $t = x$, so the formula defines nothing; negative orders need a derivative, as in [[fractional calculus|T73XK15SA2]].\nThe operators satisfy the semigroup law $I_a^\\alpha I_a^\\beta f = I_a^{\\alpha + \\beta} f$ for $\\alpha, \\beta \u003e 0$.\nFor $\\alpha = 1$, $I_a^1 f(x) = \\int_a^x f(t) \\, dt$.\n\n## Explanations\n\n### L1 human\n\nThe definition averages the past values of $f$ with the weight $(x - t)^{\\alpha - 1}$.\nWhen $\\alpha$ is a whole number, this is the usual repeated integral.\nWhen $\\alpha$ is not a whole number, for example $\\alpha = \\tfrac{1}{2}$, the same formula gives an integral of fractional order.\n",
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