{
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  "body": "\n## Statement\n\nLet $S = \\mathbb{C} \\setminus (-\\infty, 0]$ be the slit plane, the complement of the standard [[branch cut|06JYC06JHY]].\nLet $\\operatorname{Log}$ be the [[principal branch|0YKY4M3KRQ]] of the logarithm.\nThen $\\operatorname{Log}$ is [[holomorphic|0DG0Z72CQD]] on $S$, and\n$$\\operatorname{Log}'(z) = \\frac{1}{z}$$\nfor each $z \\in S$.\n\n## Remarks\n\nProof idea: on $S$, $\\operatorname{Log}$ is continuous and it is the inverse of the [[complex exponential|054VXHMCA3]] on the strip $\\{w : -\\pi \u003c \\operatorname{Im} w \u003c \\pi\\}$. The inverse function rule gives $\\operatorname{Log}'(z) = 1 / e^{\\operatorname{Log} z} = 1/z$.\n$\\operatorname{Log}$ is not continuous at the points of $(-\\infty, 0)$. For $x \u003c 0$, its values tend to $\\ln|x| + i\\pi$ from above the axis and to $\\ln|x| - i\\pi$ from below.\n\n## Explanations\n\n### L1 human\n\nAway from the cut along the negative real axis, the principal logarithm behaves like the real logarithm: its derivative is $1/z$.\nOn the cut itself, the angle jumps from about $\\pi$ to about $-\\pi$, so the function is not even continuous there.\n",
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        "url": "https://en.wikipedia.org/wiki/Complex_logarithm"
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        "locator": "The principal value as an inverse function",
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  "msc": [
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