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  "body": "\n## Definition\n\nLet $\\gamma : [a, b] \\to \\mathbb{C}$ be a closed path that is continuous and piecewise continuously differentiable. Let $w \\in \\mathbb{C}$ be a point that is not on the image of $\\gamma$.\nThe winding number of $\\gamma$ around $w$ is\n$$n(\\gamma, w) = \\frac{1}{2 \\pi i} \\oint_\\gamma \\frac{dz}{z - w},$$\nwhere the integral is a [[contour integral|00FWNP9TXY]].\n\n## Remarks\n\nThe number $n(\\gamma, w)$ is an integer. It counts how many times $\\gamma$ goes around $w$ counterclockwise, minus the times it goes around $w$ clockwise.\nAs a function of $w$, it is constant on each connected component of the complement of the image of $\\gamma$. It is $0$ on the unbounded component.\nFor the circle $|z - c| = R$, traveled once counterclockwise, $n = 1$ for $|w - c| \u003c R$ and $n = 0$ for $|w - c| \u003e R$, by the [[integral of 1/(z - w)|0SXZ68HV6F]].\n\n## Explanations\n\n### L1 human\n\nStand at the point $w$ and watch a point that moves along the closed curve. The winding number is the number of full turns that you make.\nA circle around you gives $1$, the same circle traveled twice gives $2$, and a circle that does not go around you gives $0$.\n",
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