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  "body": "\n## Summary\n\nThe Riemann–Liouville integral of order $\\alpha \u003e 0$ is the integral operator $I_a^\\alpha f(x) = \\frac{1}{\\Gamma(\\alpha)} \\int_a^x (x - t)^{\\alpha - 1} f(t) \\, dt$.\nFor a whole number $\\alpha = n$ it is the $n$-fold repeated integral, by [[Cauchy's formula for repeated integration|EYXG11V76E]].\nIt is the basic operator of [[fractional calculus|T73XK15SA2]], and fractional derivatives are defined from it.\n\n## Explanations\n\n### L1 human\n\nIntegrating $n$ times from $a$ is the same as one integral with the weight $(x - t)^{n-1} / (n - 1)!$.\nReplace $n$ by any positive real $\\alpha$ and the factorial by the [[gamma function|0H5SYBSED9]], and the formula still makes sense.\nThis gives integrals of order $1/2$, $3/2$, and so on.\nFor example, with $a = 0$, the integral of order $\\alpha$ of $1$ is $x^\\alpha / \\Gamma(\\alpha + 1)$.\n",
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