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  "body": "\n## Definition\n\nLet $s$ and $x$ be real numbers.\n1. For $s \u003e 0$ and $x \\ge 0$, the *lower incomplete gamma function* is\n$$\\gamma(s, x) = \\int_0^x t^{s-1} e^{-t}\\, dt.$$\n2. For $x \u003e 0$ and any real $s$, the *upper incomplete gamma function* is the [[improper integral|0VCTBJMEJF]]\n$$\\Gamma(s, x) = \\int_x^\\infty t^{s-1} e^{-t}\\, dt.$$\n\n## Remarks\n\nFor $s \u003e 0$ and $x \u003e 0$, $\\gamma(s, x) + \\Gamma(s, x) = \\Gamma(s)$, the [[gamma function|0H5SYBSED9]].\n\nExample: for $s = 1$, $\\gamma(1, x) = \\int_0^x e^{-t}\\, dt = 1 - e^{-x}$ and $\\Gamma(1, x) = e^{-x}$, and their sum is $1 = \\Gamma(1)$.\nIntegration by parts gives $\\gamma(s+1, x) = s\\,\\gamma(s, x) - x^s e^{-x}$.\n\nNon-example of a valid argument: for $s = 0$ the lower integral $\\int_0^x t^{-1} e^{-t}\\, dt$ diverges at $0$, so $\\gamma(0, x)$ is not defined.\nThe upper function is still defined: $\\Gamma(0, x) = E_1(x)$ is the [[exponential integral|EV6ZDQ0NDY]].\n\nThe quotient $P(s, x) = \\gamma(s, x)/\\Gamma(s)$ is the regularised lower incomplete gamma function, the distribution function of the [[gamma distribution|0B80PAQD3C]] with shape $s$ and scale $1$.\nBoth functions extend to complex arguments: $\\gamma(s, x)$ for $\\operatorname{Re} s \u003e 0$, and $\\Gamma(s, x)$ by analytic continuation.\n",
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