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  "body": "\n## Definition\n\n1. For a real number $x \u003e 0$, the exponential integral $E_1$ is the [[improper integral|05KQ3F99BQ]]\n$$E_1(x) = \\int_x^\\infty \\frac{e^{-t}}{t}\\, dt.$$\n2. For a real number $x \\ne 0$, the exponential integral $\\operatorname{Ei}$ is\n$$\\operatorname{Ei}(x) = \\int_{-\\infty}^x \\frac{e^t}{t}\\, dt,$$\nwhere for $x \u003e 0$ the integral is a [[Cauchy principal value|0SYR10HD4R]] at $t = 0$.\n\n## Remarks\n\nThe substitution $t \\mapsto -t$ gives $E_1(x) = -\\operatorname{Ei}(-x)$ for $x \u003e 0$.\nWith the [[Euler–Mascheroni constant|00C5P9JZBK]] $\\gamma$, the series\n$$\\operatorname{Ei}(x) = \\gamma + \\ln \\abs{x} + \\sum_{k=1}^\\infty \\frac{x^k}{k \\cdot k!}, \\qquad E_1(x) = -\\gamma - \\ln x - \\sum_{k=1}^\\infty \\frac{(-x)^k}{k \\cdot k!}$$\nhold for $x \\ne 0$ and $x \u003e 0$ respectively.\n\nExample: as $x \\to \\infty$, $E_1(x) \\sim \\frac{e^{-x}}{x}$, and $E_1(x) = \\Gamma(0, x)$ is the upper [[incomplete gamma function|YB7ZV0HG7K]] with $s = 0$.\n\nNon-example of a valid argument: $E_1(0)$ is not defined, since $\\int_0^1 \\frac{e^{-t}}{t}\\, dt \\ge e^{-1} \\int_0^1 \\frac{dt}{t} = \\infty$.\n\n$E_1$ extends to a holomorphic function on $\\C \\setminus (-\\infty, 0]$.\n",
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