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  "body": "\n## Summary\n\nThe complete Fermi–Dirac integral $F_j(x)$ is the integral $\\frac{1}{\\Gamma(j+1)} \\int_0^\\infty \\frac{t^j}{e^{t-x} + 1} \\, dt$.\nIt arises in the statistical physics of fermions, such as electrons in metals and semiconductors, where $x$ plays the role of a reduced chemical potential.\nIt is a [[polylogarithm|96CHDZKGQ0]] in disguise: $F_j(x) = -\\operatorname{Li}_{j+1}(-e^x)$.\n\n## Explanations\n\n### L1 human\n\nIn a gas of electrons, the chance that a state of energy $t$ is filled is $1/(e^{t - x} + 1)$, a smooth step that falls from $1$ to $0$ near $t = x$.\nWeighting this step by a power $t^j$ and integrating over all energies gives the Fermi–Dirac integral.\nFor $j = 0$ the integral can be done by hand and gives $\\log(1 + e^x)$.\nFor very negative $x$ the step is almost an exponential, and $F_j(x)$ is close to $e^x$.\n",
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