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  "body": "\n## Definition\n\nLet $a, b, c \\in \\C$, and let $w_1, w_2$ be two linearly independent solutions of the [[hypergeometric differential equation|W8VE25R5A7]] on the upper half-plane $\\mathbb{H} = \\set{z \\in \\C : \\operatorname{Im} z \u003e 0}$.\nA *Schwarz triangle function* is the ratio\n$$s(z) = \\frac{w_2(z)}{w_1(z)}, \\qquad z \\in \\mathbb{H}.$$\nWhen $c \\notin \\Z$, the standard choice uses the two solutions at $0$:\n$$s(z) = \\frac{z^{1-c} \\, {}_2F_1(a - c + 1, b - c + 1; 2 - c; z)}{{}_2F_1(a, b; c; z)},$$\nwith the principal branch of $z^{1-c}$ and the [[hypergeometric function|463X5V31D9]] ${}_2F_1$.\n\n## Remarks\n\nChanging the pair of solutions replaces $s$ by $(\\alpha s + \\beta)/(\\gamma s + \\delta)$ with $\\alpha\\delta - \\beta\\gamma \\neq 0$, so $s$ is well defined up to a Möbius transformation.\n\nLet $a, b, c$ be real and set $\\lambda = 1 - c$, $\\mu = c - a - b$, $\\nu = a - b$.\nIf $0 \\le \\lambda, \\mu, \\nu \u003c 1$, then $s$ is a [[conformal map|0YYT36R4YD]] from $\\mathbb{H}$ onto a triangle bounded by circular arcs or segments.\nIts angles are $\\pi\\lambda$, $\\pi\\mu$, $\\pi\\nu$ at the images of $0$, $1$, $\\infty$.\n\nExample: for $a = b = \\frac12$, $c = 1$ all three angles are $0$.\nHere $c \\in \\Z$, so the ratio is formed with the logarithmic second solution.\nUp to a Möbius transformation it equals $\\tau = i \\, K(1 - z)/K(z)$.\nHere $K(m) = \\frac{\\pi}{2} {}_2F_1(\\frac12, \\frac12; 1; m)$ is the complete elliptic integral of the first kind.\nIts inverse is the modular lambda function.\n\nNon-example: a ratio $w_2 / w_1$ of two proportional solutions is constant, so it is not a triangle function; independence of $w_1$ and $w_2$ is needed.\n",
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