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  "body": "\n## Definition\n\nLet $a$ and $b$ be complex numbers, let $z \\in \\C \\setminus (-\\infty, 0]$, and let $z^{1 - b}$ be the principal branch.\nLet $M$ be [[Kummer's function|6AZ4ZBN1M8]] and $\\Gamma$ the [[gamma function|0H5SYBSED9]], with $1/\\Gamma$ read as $0$ at the poles of $\\Gamma$.\nIf $b \\notin \\Z$, the *Tricomi confluent hypergeometric function* is\n$$U(a, b, z) = \\frac{\\Gamma(1 - b)}{\\Gamma(a - b + 1)} M(a, b, z) + \\frac{\\Gamma(b - 1)}{\\Gamma(a)} z^{1 - b} M(a - b + 1, 2 - b, z).$$\nIf $b \\in \\Z$, it is defined by continuity: $U(a, b, z) = \\lim_{\\beta \\to b} U(a, \\beta, z)$.\n\n## Remarks\n\nThe function $U(a, b, \\cdot)$ solves [[Kummer's equation|NT917KFR57]], since both terms of the definition do.\nIt is characterised among the solutions by $U(a, b, z) \\sim z^{-a}$ as $z \\to \\infty$ with $\\abs{\\arg z} \u003c \\frac{3\\pi}{2} - \\delta$, for each $\\delta \u003e 0$.\nFor $\\operatorname{Re} a \u003e 0$ and $\\operatorname{Re} z \u003e 0$ it has the integral form\n$$U(a, b, z) = \\frac{1}{\\Gamma(a)} \\int_0^\\infty e^{-zt}\\, t^{a - 1} (1 + t)^{b - a - 1}\\, dt.$$\nExample: $U(a, a + 1, z) = z^{-a}$; for $\\operatorname{Re} a \u003e 0$ and $\\operatorname{Re} z \u003e 0$ the integral is $\\frac{1}{\\Gamma(a)} \\int_0^\\infty e^{-zt} t^{a - 1}\\, dt = z^{-a}$.\nNon-example: $M(a, b, z)$ itself is in general not a multiple of $U(a, b, z)$; for $a = b$ it is $e^z$, which grows faster than any power as $z \\to +\\infty$.\nThe two functions satisfy Kummer's second transformation $U(a, b, z) = z^{1 - b} U(a - b + 1, 2 - b, z)$.\n",
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