{
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    "cauchys-argument-principle"
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  "body": "\n## Statement\n\nLet $c \\in \\mathbb{C}$ and $r \u003e 0$.\nLet $f$ be a [[meromorphic function|0NSD1TPHR0]] on an open set that contains the closed disc $\\{ z : |z - c| \\le r \\}$.\nSuppose that $f$ has no zeros and no poles on the circle $|z - c| = r$.\nLet $Z$ be the number of zeros of $f$ in the open disc $|z - c| \u003c r$, each counted as many times as its [[order|064N002TX4]].\nLet $P$ be the number of poles of $f$ in the open disc, each counted as many times as its [[order|0YK27TJ4WF]].\nThen\n$$\\frac{1}{2\\pi i} \\oint_{|z - c| = r} \\frac{f'(z)}{f(z)} \\, dz = Z - P,$$\nwhere the [[circle integral|0T7ABX4CWE]] goes once around the circle counterclockwise.\n\n## Remarks\n\nThe function $f'/f$ is the logarithmic derivative of $f$.\nAt a zero of order $m$, it has a simple pole with residue $m$; at a pole of order $m$, it has a simple pole with residue $-m$.\nSo the theorem follows from the [[residue theorem|032HKYS329]].\nThe left side is also the [[winding number|0X7SBS8E0F]] around $0$ of the closed curve $\\theta \\mapsto f(c + r e^{i\\theta})$, $\\theta \\in [0, 2\\pi]$.\n\n## Explanations\n\n### L1 human\n\nWalk once around the circle and watch the point $f(z)$.\nEach zero inside the circle makes $f(z)$ turn once around $0$ in the positive direction, and each pole makes it turn once in the negative direction.\nFor example, $f(z) = z^3$ on the unit circle turns three times around $0$, and $f$ has a zero of order $3$ at $0$.\n",
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    "name": "Argument principle",
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  "markdown": "https://math2.tamnd.com/node/0PN2BXRS8G.md",
  "msc": [
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  "name": "Argument principle",
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