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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\mathbb{C} \\setminus \\{0\\}$.\nA branch of the logarithm on $U$ is a continuous function $L : U \\to \\mathbb{C}$ such that $e^{L(z)} = z$ for each $z \\in U$.\nThus $L(z)$ is a [[complex logarithm|0X2FY491PF]] of $z$ for each $z \\in U$.\n\n## Remarks\n\nEach branch $L$ of the logarithm on $U$ is holomorphic on $U$, with $L'(z) = 1/z$.\nIf $U$ is a [[domain|097KNJFJ7V]] and $L_1, L_2$ are two branches on $U$, then $L_1 - L_2$ is the constant $2 \\pi i k$ for one $k \\in \\mathbb{Z}$.\nThere is no branch of the logarithm on $\\mathbb{C} \\setminus \\{0\\}$, or on any open set that contains a circle with center $0$.\nThe [[principal branch|0YKY4M3KRQ]] $\\operatorname{Log}$ is a branch on $\\mathbb{C} \\setminus (-\\infty, 0]$. The functions $\\operatorname{Log} z + 2 \\pi i k$ with $k \\in \\mathbb{Z}$ are the other branches on this set.\n\n## Explanations\n\n### L1 human\n\nThe logarithm has many values at each point, so to get a function we must choose one value at each point, and the choice must change continuously.\nSuch a choice is a branch.\nIf you go once around $0$, the argument grows by $2\\pi$, so no continuous choice is possible on a full circle around $0$.\n",
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