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  "body": "\n## Summary\n\nThe Mittag-Leffler function $E_{\\alpha, \\beta}(z) = \\sum_{k \\ge 0} \\frac{z^k}{\\Gamma(\\alpha k + \\beta)}$ generalizes the [[exponential function|0J0FCWV32D]], which is the case $\\alpha = \\beta = 1$.\nIt is an [[entire function|0G4F0XB9X2]] for $\\operatorname{Re} \\alpha \u003e 0$.\nIt plays the role in [[fractional calculus|T73XK15SA2]] that the exponential plays in ordinary calculus: it solves the basic linear fractional differential equations.\n\n## Explanations\n\n### L1 human\n\nThe exponential series is $\\sum z^k / k!$, and $k! = \\Gamma(k + 1)$.\nIf $k!$ is replaced by $\\Gamma(\\alpha k + 1)$, the result is the Mittag-Leffler function $E_\\alpha$.\nFor $\\alpha = 1$ it is $e^z$, and for $\\alpha = 2$ it is $\\cosh \\sqrt{z}$.\nFor $0 \u003c \\alpha \u003c 1$, the function $E_\\alpha(-t^\\alpha)$ decays more slowly than $e^{-t}$, like a power of $t$, which models relaxation in materials with memory.\n",
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