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  "body": "\n## Definition\n\nLet $F$ be a [[multivalued function|03CPD1HGF3]] on a set $X \\subseteq \\mathbb{C}$, so that $F(z)$ is a set of complex numbers for each $z \\in X$.\nA principal branch of $F$ is a function $f$ on $X$ with $f(z) \\in F(z)$ for each $z \\in X$, chosen by a standard convention. Its value $f(z)$ is the principal value of $F$ at $z$.\nFor the main functions of complex analysis, the conventions are these. Let $z \\neq 0$, and let $\\operatorname{Arg} z \\in (-\\pi, \\pi]$ be the [[principal value of the argument|0SQSQP748R]].\n\n- The principal [[logarithm|0X2FY491PF]] is $\\operatorname{Log} z = \\ln |z| + i \\operatorname{Arg} z$, where $\\ln$ is the real natural logarithm.\n- The principal square root is $\\sqrt{z} = e^{\\frac{1}{2} \\operatorname{Log} z}$.\n- For $w \\in \\mathbb{C}$, the principal [[power|0ZV9A7EV0F]] is $z^w = e^{w \\operatorname{Log} z}$.\n\n## Remarks\n\nThe imaginary part of $\\operatorname{Log} z$ is in $(-\\pi, \\pi]$. For example, $\\operatorname{Log} i = i\\pi/2$ and $\\operatorname{Log}(-1) = i\\pi$.\nThese principal branches are continuous and holomorphic on $\\mathbb{C} \\setminus (-\\infty, 0]$, see [[the principal logarithm is holomorphic off the branch cut|0XVGJE7Q04]]. They are not continuous at the points of $(-\\infty, 0)$.\nSome authors use $\\operatorname{Arg} z \\in [0, 2\\pi)$ instead. This gives a different principal branch.\n\n## Explanations\n\n### L1 human\n\nA multivalued function gives many values at a point. The principal branch is the rule that everybody agrees on for picking one value.\nFor the logarithm, the rule is to take the angle between $-180$ and $180$ degrees.\n",
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