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  "body": "\n## Statement\n\nLet $\\gamma : [a, b] \\to \\mathbb{C}$ be a continuous and piecewise continuously differentiable path with length $L(\\gamma) = \\int_a^b |\\gamma'(t)| \\, dt$.\nLet $f$ be a complex function that is continuous on the image of $\\gamma$.\nLet $M \\ge 0$ be a real number with $|f(z)| \\le M$ for each $z$ on the image of $\\gamma$.\nThen the [[contour integral|00FWNP9TXY]] of $f$ along $\\gamma$ satisfies\n$$\\left| \\int_\\gamma f(z) \\, dz \\right| \\le M \\, L(\\gamma).$$\n\n## Remarks\n\nFor a [[circle integral|0T7ABX4CWE]] over $|z - c| = R$, the bound is $\\left| \\oint f(z) \\, dz \\right| \\le 2 \\pi R M$.\nProof: $\\left| \\int_a^b g(t) \\, dt \\right| \\le \\int_a^b |g(t)| \\, dt$ for a complex function $g$, and $|f(\\gamma(t)) \\gamma'(t)| \\le M |\\gamma'(t)|$.\nA typical use shows that the integral over a large half-circle goes to $0$ as the radius goes to $\\infty$.\n\n## Explanations\n\n### L1 human\n\nThe integral cannot be larger than the largest size of the function times the length of the path.\nFor example, if $|f| \\le 1$ on a circle of radius $3$, then the integral over the circle has size at most $6\\pi$.\n",
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    "name": "Estimation lemma",
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  "name": "Estimation lemma",
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