lectures:gaussian_integral
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| lectures:gaussian_integral [2022/01/07 14:02] – kimi | lectures:gaussian_integral [2022/08/23 13:34] (現在) – 外部編集 127.0.0.1 | ||
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| ====== Gaussian integral ====== | ====== Gaussian integral ====== | ||
| - | + | ===== 計算例1 ===== | |
| $$ | $$ | ||
| \begin{align} | \begin{align} | ||
| - | \int_{0}^{\infty}e^{-(1+t^2)x}dx&=\left[-\frac{e^{-(1+t^2)x}}{1+t^2}\right]_{0}^{\infty}=\frac{1}{1+t^2}\\ | + | I&=\int_{-\infty}^{\infty}e^{-x^2}dx\\ |
| - | \int_{0}^{\infty}\int_{0}^{\infty}e^{-(1+t^2)x}dxdt& | + | &=\int_{-\infty}^{\infty}e^{-y^2}dy\\ |
| I^2& | I^2& | ||
| & | & | ||
| 行 17: | 行 16: | ||
| $$ | $$ | ||
| + | $I=\sqrt{\pi}$ | ||
| - | --- | + | ===== 計算例2 ===== |
| $$ | $$ | ||
| \begin{align} | \begin{align} | ||
| - | I& | + | \frac{I}{2}& |
| - | &=\int_{-\infty}^{\infty}e^{-y^2}dy\\ | + | \int_{0}^{\infty}e^{-(1+t^2)x}dx&=\left[-\frac{e^{-(1+t^2)x}}{1+t^2}\right]_{0}^{\infty}=\frac{1}{1+t^2}\\ |
| - | I^2&=\int_{-\infty}^{\infty}e^{-x^2}dx\int_{-\infty}^{\infty}e^{-y^2}dy\\ | + | \int_{0}^{\infty}\int_{0}^{\infty}e^{-(1+t^2)x}dxdt&=\int_{0}^{\infty}\frac{1}{1+t^2}dt\\ |
| - | &=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dxdy\\ | + | \int_{0}^{\infty}e^{-x}\int_{0}^{\infty}e^{-xt^2}dtdx& |
| - | &=\int_{0}^{2\pi}\int_{0}^{\infty}e^{-r^2}rdrd\theta\\ | + | \int_{0}^{\infty}e^{-x}\frac{1}{\sqrt{x}}\int_{0}^{\infty}e^{-\tau^2}d\tau dx& |
| - | &=2\pi\int_{0}^{\infty}e^{-r^2}rdr\\ | + | \int_{0}^{\infty}\frac{e^{-x}}{\sqrt{x}}dx\int_{0}^{\infty}e^{-\tau^2}d\tau &=\frac{\pi}{2}\\ |
| - | & | + | \int_{0}^{\infty}\frac{e^{-u^2}}{u}2udu\int_{0}^{\infty}e^{-\tau^2}d\tau & |
| - | &=\pi\\ | + | 2\frac{I}{2}\frac{I}{2}& |
| + | I^2&=\pi\\ | ||
| I& | I& | ||
| \end{align} | \end{align} | ||
| $$ | $$ | ||
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| + | --- | ||
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lectures/gaussian_integral.1641531729.txt.gz · 最終更新: 2022/08/23 13:34 (外部編集)