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  "body": "\n## Definition\n\nLet $a, b, c \\in \\C$ be distinct, and let $\\alpha, \\alpha', \\beta, \\beta', \\gamma, \\gamma' \\in \\C$ satisfy\n$$\\alpha + \\alpha' + \\beta + \\beta' + \\gamma + \\gamma' = 1.$$\n*Riemann's differential equation* with these data is the [[ordinary differential equation|0NT64JV96A]]\n$$\\begin{aligned} w'' \u0026+ \\left[\\frac{1 - \\alpha - \\alpha'}{z - a} + \\frac{1 - \\beta - \\beta'}{z - b} + \\frac{1 - \\gamma - \\gamma'}{z - c}\\right] w' \\\\ \u0026+ \\left[\\frac{\\alpha\\alpha'(a - b)(a - c)}{z - a} + \\frac{\\beta\\beta'(b - c)(b - a)}{z - b} + \\frac{\\gamma\\gamma'(c - a)(c - b)}{z - c}\\right] \\frac{w}{(z - a)(z - b)(z - c)} = 0. \\end{aligned}$$\n\n## Remarks\n\nIts singular points are $a$, $b$, $c$, all [[regular singular points|H96HXG6CA0]], and $\\infty$ is an ordinary point.\nThe exponents are $\\alpha, \\alpha'$ at $a$, $\\beta, \\beta'$ at $b$ and $\\gamma, \\gamma'$ at $c$.\n\nCheck at $z = a$: multiplying by $(z - a)^2$ and letting $z \\to a$, the indicial equation is $r(r - 1) + (1 - \\alpha - \\alpha') r + \\alpha\\alpha' = (r - \\alpha)(r - \\alpha') = 0$, so the exponents at $a$ are $\\alpha$ and $\\alpha'$.\n\nExample: take the singular points $0$, $\\infty$, $1$, using the limiting form in which the factors with $b$ are dropped as $b \\to \\infty$.\nWith exponents $0, 1 - C$ at $0$, $A, B$ at $\\infty$ and $0, C - A - B$ at $1$, the equation becomes the [[hypergeometric differential equation|W8VE25R5A7]] with parameters $A, B, C$.\n\nNon-example: exponents with sum different from $1$ are not allowed, because the equation then acquires a fourth singular point at $\\infty$.\nThe condition on the sum is Fuchs's relation for three singular points.\n\nThe solutions are written with the [[Riemann P-symbol|WWW053WXP2]].\n",
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