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  "body": "\n## Definition\n\nLet $a, b, c \\in \\C$, let $X = \\C \\setminus \\set{0, 1}$, and fix a base point $z_0 \\in X$.\nLet $V$ be the two-dimensional complex vector space of solutions of the [[hypergeometric differential equation|W8VE25R5A7]] on a small disc around $z_0$.\nFor a loop $\\gamma$ in $X$ based at $z_0$, [[analytic continuation|0M3YQJ81NB]] of each solution along $\\gamma$ gives a solution near $z_0$ again.\nThis defines a linear map $M_\\gamma \\colon V \\to V$ that depends only on the homotopy class of $\\gamma$.\nThe *monodromy representation* of the equation is the [[group representation|WYGRXCG2RN]]\n$$\\rho \\colon \\pi_1(X, z_0) \\to \\mathrm{GL}(V), \\qquad [\\gamma] \\mapsto M_\\gamma,$$\nof the [[fundamental group|S1XK3HXHFX]], with the convention that makes $\\rho$ a homomorphism.\nIts image is the *monodromy group* of the equation, defined up to conjugation in $\\mathrm{GL}_2(\\C)$.\n\n## Remarks\n\nThe group $\\pi_1(X, z_0)$ is free on the classes $\\gamma_0$, $\\gamma_1$ of simple loops around $0$ and $1$, so $\\rho$ is determined by the two matrices $M_{\\gamma_0}$ and $M_{\\gamma_1}$.\nThe eigenvalues of $M_{\\gamma_0}$ are $1$ and $e^{2\\pi i(1 - c)}$, and those of $M_{\\gamma_1}$ are $1$ and $e^{2\\pi i(c - a - b)}$.\nA loop around $\\infty$, positively oriented in the coordinate $1/z$, has eigenvalues $e^{2\\pi i a}$ and $e^{2\\pi i b}$.\n\nExample: for $a = b = 1$, $c = 2$, the solutions near $z_0$ are spanned by $w_1 = 1/z$ and $w_2 = \\log(1 - z)/z$, as a direct check in the equation shows.\nBoth are single-valued near $0$, so $M_{\\gamma_0}$ is the identity.\nAround $1$, $\\log(1 - z)$ changes by $\\pm 2\\pi i$, so in the basis $(w_1, w_2)$ the matrix $M_{\\gamma_1}$ is $\\begin{pmatrix} 1 \u0026 \\pm 2\\pi i \\\\ 0 \u0026 1 \\end{pmatrix}$, with the sign fixed by the orientation.\nHere the monodromy group is infinite cyclic.\n\nNon-example: continuation along a path from $z_0$ to a different point $z_1$ does not give an element of the monodromy group.\nIt maps solutions near $z_0$ to solutions near $z_1$, which is a different space.\n\nThis is a special case of the general notion of [[monodromy|C8HD62AB6Q]] of a linear differential equation.\n",
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