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  "body": "\n## Summary\n\nThe Gauss hypergeometric function ${}_2F_1(a, b; c; z)$ is the sum of the [[hypergeometric series|428M453CR2]] on the unit disc.\nIt is extended by [[analytic continuation|0M3YQJ81NB]] to the cut plane $\\C \\setminus [1, \\infty)$.\nIt solves the [[hypergeometric differential equation|W8VE25R5A7]], to which every second-order linear equation with three regular singular points can be reduced.\nMost classical special functions, such as Legendre functions, complete elliptic integrals, Jacobi polynomials and many elementary functions, are special cases or limits of it.\n\n## Explanations\n\n### L1 human\n\nThe hypergeometric function is a single formula with three parameters $a, b, c$ that produces a large family of familiar functions.\nFor different choices of the parameters it gives powers $(1-z)^{-a}$, logarithms, inverse sines, and orthogonal polynomials.\nThe power series only converges for $\\abs{z} \u003c 1$, but the function can be extended to all complex $z$ except a cut along $[1, \\infty)$.\nFor example, ${}_2F_1(\\tfrac12, \\tfrac12; \\tfrac32; z^2) = \\frac{\\arcsin z}{z}$ for $0 \u003c \\abs{z} \u003c 1$.\nBecause so many functions are special cases, a single identity for ${}_2F_1$ often gives many classical identities at once.\n",
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