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m Change the notation for multiple integral |
→Gaussian integral: added some details |
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In physics the factor of 1/2 in the argument of the exponential is common.
Note that, if we let <math>r=\sqrt{x^2+y^2}</math> be the radius, then we can use the usual polar coordinate change of variables (which in particular renders <math>dx\,dy=r\,dr\,d\theta</math>) to get
<math display="block"> G^2 = \left ( \int_{-\infty}^{\infty} e^{-{1 \over 2} x^2}\,dx \right ) \cdot \left ( \int_{-\infty}^{\infty} e^{-{1 \over 2} y^2}\,dy \right ) = 2\pi \int_{0}^{\infty} r e^{-{1 \over 2} r^2}\,dr = 2\pi \int_{0}^{\infty} e^{- w}\,dw = 2 \pi.</math>
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