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  "body": "\n## Summary\n\nThe Tricomi confluent hypergeometric function $U(a, b, z)$ is the solution of [[Kummer's equation|NT917KFR57]] that behaves like $z^{-a}$ as $z \\to +\\infty$.\nIt is a combination of two Kummer functions $M$.\nIt grows or decays only like a power of $z$ at infinity, while $M(a, b, z)$ in general grows like $e^z$.\nMany functions of applied mathematics are special cases of $U$.\nExamples are the complementary error function, the upper incomplete gamma function and the modified Bessel function $K_\\nu$.\n\n## Explanations\n\n### L1 human\n\nKummer's equation has a two-dimensional space of solutions.\nKummer's function $M$ is the solution that is nice at $z = 0$.\nTricomi's function $U$ is the solution that is nice at $z = +\\infty$: it behaves like a power of $z$ instead of growing exponentially.\nFor $\\operatorname{Re} a \u003e 0$ and $\\operatorname{Re} z \u003e 0$ it has the integral form $U(a, b, z) = \\frac{1}{\\Gamma(a)} \\int_0^\\infty e^{-zt}\\, t^{a - 1} (1 + t)^{b - a - 1}\\, dt$.\nFor $b = a + 1$ the factor $(1 + t)^{b - a - 1}$ is $1$, and the integral gives $U(a, a + 1, z) = z^{-a}$.\n",
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