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  "body": "\n## Definition\n\nLet $a \\in \\C$ and $r \u003e 0$, and let $D^* = \\set{z : 0 \u003c \\abs{z - a} \u003c r}$ be a punctured disc.\nLet $F$ be a [[multivalued function|03CPD1HGF3]] on $D^*$ given by a [[holomorphic|0DG0Z72CQD]] function element that admits [[analytic continuation|0M3YQJ81NB]] along every path in $D^*$.\nThe point $a$ is a *branch point* of $F$ if some function element of $F$ changes after continuation once around a circle $\\abs{z - a} = \\rho$ with $0 \u003c \\rho \u003c r$.\nThe point $\\infty$ of the [[Riemann sphere|0CDMXGPSES]] is a branch point of $F$ if $0$ is a branch point of $w \\mapsto F(1/w)$.\n\n## Remarks\n\nIf $n$ turns around $a$ bring each element back to itself for some $n \\ge 2$, the branch point is *algebraic*, and the least such $n$ is its order.\nOtherwise $a$ is a *transcendental* branch point; a *logarithmic* branch point, where no number of turns brings any element back, is the main example.\nSome authors call $n - 1$ the order instead of $n$.\nExample: for $F(z) = \\sqrt{z}$, continuing the element $e^{(\\log z)/2}$ once around $\\abs{z} = 1$ multiplies it by $e^{i\\pi} = -1$.\nSo $0$ is an algebraic branch point of order $2$, and so is $\\infty$, because $\\sqrt{1/w}$ also changes sign around $w = 0$.\nThe [[complex logarithm|02QGGD5G87]] has logarithmic branch points at $0$ and $\\infty$, since each turn adds $2\\pi i$.\nNon-example: $0$ is not a branch point of $1/z$, which is single-valued on $\\C \\setminus \\set{0}$; it is a pole.\nFor a holomorphic map between [[Riemann surfaces|5K0N8QMV6Q]], a point where the map is not locally injective is also called a branch point or ramification point.\n",
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