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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{C}$ be an open set and let $F$ be a [[multivalued function|03CPD1HGF3]] on $U$.\nA branch cut for $F$ is a closed subset $K$ of $U$ such that $F$ has a continuous single-valued branch on the open set $U \\setminus K$. Usually a branch cut is a curve or a union of curves.\nA branch on $U \\setminus K$ is a continuous function $f$ on $U \\setminus K$ with $f(z) \\in F(z)$ for each $z \\in U \\setminus K$.\nThe standard branch cut for the logarithm on $\\mathbb{C} \\setminus \\{0\\}$ is the ray $(-\\infty, 0)$. The open set $\\mathbb{C} \\setminus (-\\infty, 0]$ is the slit plane.\n\n## Remarks\n\nA branch cut usually joins branch points. A branch point of $F$ is a point $z_0$ with the following property. When $z$ goes once around a small circle with center $z_0$, the values of $F$ do not come back to their start.\nFor $\\log z$ and $\\sqrt z$, the branch points are $0$ and $\\infty$, so a cut is a curve from $0$ to $\\infty$.\nThe [[principal branches|0YKY4M3KRQ]] of the logarithm and of the square root use the slit plane. Each other ray from $0$ is also a valid branch cut and gives other [[branches|0SQ1F2BEDS]].\nThe slit plane is $\\{z \\in \\mathbb{C} : \\operatorname{Re} z \u003e 0 \\text{ or } \\operatorname{Im} z \\neq 0\\}$.\n\n## Explanations\n\n### L1 human\n\nYou cannot choose the logarithm continuously on a full circle around $0$.\nA branch cut is a line that you remove from the plane so that no path can go around $0$. On what is left, a continuous choice is possible.\nCrossing the cut makes the chosen value jump, for the logarithm by $2\\pi i$.\n",
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