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  "body": "\n## Statement\n\nLet $a, b, c \\in \\C$ with $c \\notin \\set{0, -1, -2, \\ldots}$, and let ${}_2F_1$ be the [[hypergeometric function|463X5V31D9]] on $\\C \\setminus [1, \\infty)$, with principal powers.\nThen for all $z \\in \\C \\setminus [1, \\infty)$:\n\n1. (Pfaff transformations)\n$${}_2F_1(a, b; c; z) = (1 - z)^{-a} \\, {}_2F_1\\left(a, c - b; c; \\frac{z}{z - 1}\\right) = (1 - z)^{-b} \\, {}_2F_1\\left(c - a, b; c; \\frac{z}{z - 1}\\right).$$\n2. (Euler transformation)\n$${}_2F_1(a, b; c; z) = (1 - z)^{c - a - b} \\, {}_2F_1(c - a, c - b; c; z).$$\n\n## Remarks\n\nThe map $z \\mapsto \\frac{z}{z - 1}$ sends $\\C \\setminus [1, \\infty)$ onto itself, so the right sides make sense.\nFor $\\operatorname{Re} c \u003e \\operatorname{Re} b \u003e 0$, the first Pfaff formula follows from [[Euler's integral representation|812THX456R]] by the substitution $t \\mapsto 1 - t$, and the general case by analytic continuation in the parameters.\nThe second Pfaff formula is the first with $a$ and $b$ exchanged, and applying the two Pfaff formulas one after the other gives the Euler transformation.\n\nExample: for $b = c$, the Euler transformation reads $(1 - z)^{-a} = (1 - z)^{-a} \\, {}_2F_1(c - a, 0; c; z)$, which is true since ${}_2F_1(\\cdot, 0; c; z) = 1$.\n\nThe Pfaff transformation turns the series at $z$ into a series at $\\frac{z}{z - 1}$, which converges for $\\operatorname{Re} z \u003c \\frac12$; this gives the continuation of ${}_2F_1$ to that half-plane.\n\n## Explanations\n\n### L1 human\n\nThe same hypergeometric function can be written in several different ways, with changed parameters, a changed variable, and a power of $1 - z$ in front.\nThese formulas let one pass from a series that converges slowly to one that converges fast, or to a region where the original series does not converge at all.\n\n### L2 human\n\nTogether with the identity, the Pfaff and Euler transformations form a group of four expressions for the same solution of the hypergeometric equation.\nCombined with the solutions at the other singular points, they give [[Kummer's 24 solutions|7QFW771QMA]].\nThe Euler transformation also explains the factor $(1 - z)^{c - a - b}$ that governs the behaviour of ${}_2F_1$ near $z = 1$.\n",
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