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注记$\displaystyle\;\,0={\small\oint}_{\,\Gamma_{(r,R)}}{\small\frac{e^{iz}}{z}}dz={\small\int_{-R}^{-r}\frac{e^{ix}}{x}}dx+{\small\int_{\pi}^0\frac{\exp(ire^{i\theta})}{z}}izd\theta$:9w $\qquad\qquad\qquad\displaystyle\qquad\quad{\small+\int_r^R\frac{e^{ix}}{x}}dx+{\small\int_0^{\pi}\frac{\exp(iRe^{i\theta})}{z}}izd\theta$>7[} N $\qquad\;=\displaystyle{\small\int_r^R\frac{e^{ix}-e^{-ix}}{x}}dx+i{\small\int_0^{\pi}(e^{-R\sin\theta}e^{iR\cos\theta}-\exp(rie^{i\theta}))d\theta}$:kB $\qquad\longrightarrow 2i{\small\displaystyle\int_0^{\infty}\frac{\sin x}{x}}dx+i0-i\pi\;(r,R^{-1}\to 0)$"B 其中闭路径$\;\;\small\underset{\,}{\Gamma_{(r,R)}}=[-R,-r]\oplus\text{Arc}(r,\pi,0)\oplus[r,R]\oplus\text{Arc}(R,0,\pi).$*P,jve
$\therefore\;\;\displaystyle{\small\int_{-\infty}^{\infty}\frac{\sin x}{x}}dx=\pi$}NUY5
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