Thread: Gaussian integral trouble View Single Post
 P: 98 OK, I figured it out. The starting point was supposed to be: $$\int_{-\infty}^{\infty} dx.x^2.e^{-\frac{1}{2}ax^2} = a^{-1}\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2}$$ so I'd get to: $$\int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = -2 \{ \frac{\partial a^{-1}}{\partial a}.\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} + a^{-1}.\frac{\partial}{\partial a}\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} \}$$ $$\int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = -2 \{ -a^{-2}.\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} + a^{-1}.\int_{-\infty}^{\infty} dx.\frac{\partial}{\partial a}e^{-\frac{1}{2}ax^2} \}$$ $$\int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = -2 \{ -a^{-2}.\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} + a^{-1}.-\frac{1}{2}.\int_{-\infty}^{\infty} dx.x^2.e^{-\frac{1}{2}ax^2} \}$$ $$\int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = -2 \{ -a^{-2}.\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} + a^{-1}.-\frac{1}{2}.a^{-1}\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} \}$$ $$= 3a^{-2} \int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2}$$ as expected. Now for that funky matrix stuff, which will doubtless lead me back here.