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  "body": "\n## Statement\n\nLet $a, b, c \\in \\C$ with $\\operatorname{Re} c \u003e \\operatorname{Re} b \u003e 0$, and let $\\Gamma$ be the [[gamma function|0H5SYBSED9]].\nThen for every $z \\in \\C \\setminus [1, \\infty)$,\n$${}_2F_1(a, b; c; z) = \\frac{\\Gamma(c)}{\\Gamma(b) \\, \\Gamma(c - b)} \\int_0^1 t^{b-1} (1 - t)^{c - b - 1} (1 - zt)^{-a} \\, dt,$$\nwhere ${}_2F_1$ is the [[hypergeometric function|463X5V31D9]] and $(1 - zt)^{-a}$ is the principal power.\n\n## Remarks\n\nFor $\\abs{z} \u003c 1$, expand $(1 - zt)^{-a} = \\sum_n \\frac{(a)_n}{n!} z^n t^n$ by the binomial series and integrate term by term.\nThe $n$-th term gives the [[beta function|QHNWVSC6AQ]] value $B(b + n, c - b) = \\frac{\\Gamma(b + n) \\Gamma(c - b)}{\\Gamma(c + n)}$, and $\\Gamma(b + n)/\\Gamma(b) = (b)_n$, $\\Gamma(c + n)/\\Gamma(c) = (c)_n$, which gives the series of ${}_2F_1$.\nThe integral is holomorphic in $z \\in \\C \\setminus [1, \\infty)$, because $1 - zt$ never lies on $(-\\infty, 0]$ there, so the identity extends by analytic continuation.\n\nExample: with $a = 1$, $b = 1$, $c = 2$ the integral is $\\int_0^1 \\frac{dt}{1 - zt} = -\\frac{\\log(1 - z)}{z}$, which is ${}_2F_1(1, 1; 2; z)$.\n\nThe hypothesis $\\operatorname{Re} c \u003e \\operatorname{Re} b \u003e 0$ is needed for convergence of the integral at $t = 0$ and $t = 1$.\nFor $b = c$ the factor $(1 - t)^{-1}$ makes the integral diverge at $t = 1$, although ${}_2F_1(a, b; b; z) = (1 - z)^{-a}$ is perfectly defined.\nA Pochhammer contour integral removes this restriction.\n\n## Explanations\n\n### L1 human\n\nThe hypergeometric function is first defined by a power series that only converges for $\\abs{z} \u003c 1$.\nEuler's formula writes it as an integral over $[0, 1]$ of an explicit function.\nThe integral makes sense for every $z$ off the cut $[1, \\infty)$, so it shows directly how to extend the function there.\n\n### L2 human\n\nThis representation is the main tool for proving identities for ${}_2F_1$.\nSetting $z = 1$ gives [[Gauss's hypergeometric theorem|1FC0T0GCCX]] by the beta integral, and the substitution $t \\mapsto 1 - t$ gives the [[Pfaff transformation|KAAJ0JJ2GF]].\nIt also exhibits ${}_2F_1$ as a period integral of a twisted differential form, which leads to the monodromy and to generalizations such as Appell and Lauricella functions.\n",
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