高斯積分

高斯積分係高斯函數（e^−x2）喺整個實數線上面嘅積分：

${\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}dx={\sqrt {\pi }}}$

證明

用平面坐標同極坐標嘅轉換可以計到高斯積分。

設 ${\displaystyle I=\int _{-\infty }^{\infty }e^{-x^{2}}dx}$。

${\displaystyle I^{2}}$

${\displaystyle =(\int _{-\infty }^{\infty }e^{-x^{2}}dx)^{2}}$

${\displaystyle =(\int _{-\infty }^{\infty }e^{-x^{2}}dx)(\int _{-\infty }^{\infty }e^{-y^{2}}dy)}$

${\displaystyle =\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{x^{2}+y^{2}}dxdy}$

${\displaystyle =\int _{0}^{2\pi }\int _{0}^{\infty }re^{-r^{2}}drd\theta }$ （坐標轉換）

${\displaystyle =\int _{0}^{2\pi }\int _{0}^{\infty }e^{-u}dud\theta }$

${\displaystyle =\int _{0}^{2\pi }-{\frac {e^{-u}}{2}}{\big |}_{0}^{\infty }d\theta }$

${\displaystyle =\int _{0}^{2\pi }{\frac {d\theta }{2}}}$

${\displaystyle ={\frac {\theta }{2}}{\big |}_{0}^{2\pi }}$

${\displaystyle =\pi }$

${\displaystyle \therefore I={\sqrt {\pi }}}$

應用

喺統計學嘅正態分佈裏面，佢嘅概率密度函數公式係 ${\displaystyle {\frac {1}{{\sqrt {2\pi }}\sigma }}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}$。利用以上嘅方法可以證明呢個函數喺 ${\displaystyle (-\infty ,\infty )}$ 嘅積分係 ${\displaystyle 1}$。