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+$I^2=\int_0^{+\infty}e^{-x^2}\,\textrm{d}x\cdot\int_0^{+\infty}e^{-x^2}\,\textrm{d}x=\int_0^{+\infty}e^{-x^2}\,\textrm{d}x\cdot\int_0^{+\infty}e^{-y^2}\,\textrm{d}y$ + +$=\displaystyle{\iint\limits_{\substack{0\leqslant x\leqslant+\infty\\0\leqslant y\leqslant+\infty}}e^{-(x^2+y^2)}\,\textrm{d}x\textrm{d}y}$,令$x=r\cos\theta$,$y=r\sin\theta$: + +$=\displaystyle{\int_0^\frac{\pi}{2}\textrm{d}\theta\int_0^{+\infty}e^{-r^2}r\,\textrm{d}r=\dfrac{\pi}{2}\left(-\dfrac{1}{2}\right)\int_0^{+\infty}e^{-r^2}\,\textrm{d}(-r^2)=-\dfrac{\pi}{4}e^{-r^2}\bigg\vert_0^{+\infty}}=\dfrac{\pi}{4}$。 + +$\therefore I=\dfrac{\sqrt{\pi}}{2}$。 + \end{document} diff --git a/advanced-math/knowledge/8-infinite-series/infinite-series.pdf b/advanced-math/knowledge/8-infinite-series/infinite-series.pdf index e59202181dbacdd06cc380cfde819477325e7660..d430d119ab5b3ebfe1b2a566a313292e4b17bf13 100644 GIT binary patch delta 86197 zcmY(qQ*@y1?=?KNr?xq@ZQHi(c52;I+qN~eZQIt=w*5Ze@96uV!NRl5< 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