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  "body": "\n## Statement\n\nLet $a \u003e 0$ and $R \u003e 0$ be real numbers.\nLet $C_R$ be the semicircle $\\{ R e^{i\\theta} : \\theta \\in [0, \\pi] \\}$ in the upper half-plane, with the parametrization $\\theta \\mapsto R e^{i\\theta}$.\nLet $g$ be a continuous complex function on $C_R$, and let $f(z) = e^{iaz} g(z)$ for $z \\in C_R$, where $e^w$ is the [[complex exponential|054VXHMCA3]].\nLet $M_R = \\max_{\\theta \\in [0, \\pi]} |g(R e^{i\\theta})|$.\nThen the [[contour integral|00FWNP9TXY]] of $f$ along $C_R$ satisfies\n$$\\left| \\int_{C_R} f(z) \\, dz \\right| \\le \\frac{\\pi}{a} M_R.$$\n\n## Remarks\n\nIf $M_R \\to 0$ as $R \\to \\infty$, then $\\int_{C_R} f(z) \\, dz \\to 0$ as $R \\to \\infty$.\nThe bound does not contain the length $\\pi R$ of $C_R$, so it is stronger than the [[estimation lemma|0DJMGPN8K9]] for this type of integrand.\nA similar result holds for the semicircle in the lower half-plane when $a \u003c 0$.\nThe lemma is a main step in the [[evaluation of real integrals|0476QH9846]] such as $\\int_{-\\infty}^{\\infty} \\frac{e^{iax}}{x^2 + 1} \\, dx$.\n\n## Explanations\n\n### L1 human\n\nOn the semicircle, $|e^{iaz}| = e^{-aR \\sin \\theta}$, and this number is very small except near the two ends of the semicircle.\nSo the factor $e^{iaz}$ kills most of the long path.\nThis is why the bound stays small even when the radius $R$ is large.\n",
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