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  "body": "\n## Summary\n\nThe exponential integrals are the functions $E_1(x) = \\int_x^\\infty \\frac{e^{-t}}{t}\\, dt$ for $x \u003e 0$ and $\\operatorname{Ei}(x)$, the principal value of $\\int_{-\\infty}^x \\frac{e^t}{t}\\, dt$ for $x \\ne 0$.\nThey are related by $E_1(x) = -\\operatorname{Ei}(-x)$ for $x \u003e 0$, and $E_1(x)$ is the [[upper incomplete gamma function|YB7ZV0HG7K]] $\\Gamma(0, x)$.\nThe [[logarithmic integral|VP98WSTZWT]] is $\\operatorname{li}(x) = \\operatorname{Ei}(\\ln x)$.\nThey occur in heat transfer, in groundwater flow and in radiative transfer.\n\n## Explanations\n\n### L1 human\n\nThe function $e^t / t$ has no antiderivative made from elementary functions.\nThe exponential integral gives a name to its antiderivative, fixed by a choice of starting point.\nThe function $E_1(x)$ is the area under $e^{-t}/t$ to the right of $x$; it is large near $0$ and very small for large $x$.\nNear $0$ it behaves like $-\\ln x$, because $e^{-t}/t$ is close to $1/t$ there.\n",
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