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  "body": "\n## Definition\n\nLet $a, b, c \\in \\C$ with $c \\notin \\set{0, -1, -2, \\ldots}$.\nThe *hypergeometric function* ${}_2F_1(a, b; c; \\cdot)$ is the unique [[holomorphic function|0NAHXR1FRH]] on $\\C \\setminus [1, \\infty)$ that agrees on the disc $\\abs{z} \u003c 1$ with the sum of the [[hypergeometric series|428M453CR2]]\n$$\\sum_{n=0}^{\\infty} \\frac{(a)_n (b)_n}{(c)_n} \\, \\frac{z^n}{n!}.$$\n\n## Remarks\n\nExistence of this [[analytic continuation|0M3YQJ81NB]] follows from Euler's integral representation when $\\operatorname{Re} c \u003e \\operatorname{Re} b \u003e 0$.\nIn general it follows from the hypergeometric differential equation, whose only finite singular points are $0$ and $1$.\nUniqueness holds because $\\C \\setminus [1, \\infty)$ is connected.\nThis is the principal branch; continuation across the cut gives other branches of a multivalued function.\n\nExample: ${}_2F_1(a, b; b; z) = (1 - z)^{-a}$ on the whole cut plane, with the principal power, since both sides are holomorphic there and agree for $\\abs{z} \u003c 1$.\n\nNon-example: ${}_2F_1(1, 1; 2; z) = -\\log(1 - z)/z$ has no holomorphic extension to a neighbourhood of $z = 1$ or to any annulus around $1$: going once around $z = 1$ changes $\\log(1 - z)$ by $2\\pi i$.\nSo the cut, or a choice of branch, is needed.\n\nGauss's notation $F(a, b; c; z)$ is common, and the normalized function ${}_2F_1(a, b; c; z)/\\Gamma(c)$ is entire in $c$.\n",
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