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  "body": "\n## Summary\n\nFractional calculus studies integrals and [[derivatives|0V1EJCT8G6]] of non-integer order, such as a \"half derivative\".\nIts basic operator is the [[Riemann–Liouville integral|JTKCZ44NN3]], which extends repeated integration to any real order $\\alpha \u003e 0$.\nFractional derivatives, such as the Riemann–Liouville, [[Caputo|6VMGQYF8RW]] and [[Grünwald–Letnikov|M9Z8VM4K1Z]] derivatives, are built from it or from limits of differences.\nThese operators model memory effects and anomalous diffusion, and appear in integral equations such as Abel's equation.\n\n## Explanations\n\n### L1 human\n\nDifferentiating once, twice or three times is familiar; fractional calculus asks what it means to differentiate half a time.\nThe idea is to find a formula for the $n$-th integral that makes sense when $n$ is not a whole number.\n[[Cauchy's formula for repeated integration|EYXG11V76E]] gives such a formula, with the factorial replaced by the [[gamma function|0H5SYBSED9]].\nFor example, the half integral of $1$ from $0$ is $2\\sqrt{x/\\pi}$, and doing the half integral twice gives the ordinary integral $x$.\n",
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