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  "body": "\n## Definition\n\nLet $a \\in \\R$ and let $f$ be a locally integrable real function on $[a, \\infty)$.\nFor a real number $\\alpha \u003e 0$, the fractional integral of order $\\alpha$ of $f$ is the [[Riemann–Liouville integral|JTKCZ44NN3]]\n$$I_a^\\alpha f(x) = \\frac{1}{\\Gamma(\\alpha)} \\int_a^x (x - t)^{\\alpha - 1} f(t) \\, dt, \\qquad x \u003e a,$$\nwhere $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\nFor $\\alpha \u003e 0$ let $n = \\ceil{\\alpha}$.\nThe Riemann–Liouville fractional derivative of order $\\alpha$ is\n$$D_a^\\alpha f(x) = \\frac{d^n}{dx^n} \\, I_a^{n - \\alpha} f(x),$$\nwhenever the right side exists, with $I_a^0 f = f$.\n*Fractional calculus* is the study of these operators and of their variants, for real or complex orders $\\alpha$.\n\n## Remarks\n\nFor a positive integer $\\alpha = n$, $I_a^n f$ is the $n$-fold iterated integral of $f$ from $a$, and $D_a^n f$ is the ordinary [[higher derivative|04TKTF8KDK]] $f^{(n)}$.\nExample: with $a = 0$ and $f(t) = 1$, $I_0^{1/2} f(x) = \\frac{1}{\\Gamma(1/2)} \\int_0^x (x - t)^{-1/2} \\, dt = \\frac{2\\sqrt{x}}{\\sqrt{\\pi}}$.\nNon-example: for $\\alpha = 0$ the integral formula is not defined, since $\\Gamma(0)$ is not defined; the operator $I_a^0$ is set to be the identity by convention.\nThe fractional integrals satisfy $I_a^\\alpha I_a^\\beta = I_a^{\\alpha + \\beta}$ for $\\alpha, \\beta \u003e 0$.\nOther fractional derivatives, such as the [[Caputo derivative|6VMGQYF8RW]] and the [[Grünwald–Letnikov derivative|M9Z8VM4K1Z]], agree with $D_a^\\alpha$ on suitable classes of functions.\n\n## Explanations\n\n### L1 human\n\nThe fractional integral replaces \"integrate $n$ times\" by a single weighted integral whose weight makes sense for any positive order.\nA fractional derivative of order $\\alpha$ is then obtained by integrating a little, to order $n - \\alpha$, and differentiating a whole number $n$ of times.\n",
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