Even Powers

The Gauss integral shows up in statistical mechanics, in kinetic theory of gases, and in elementary statistics, and related integrals are often required. Define the Gauss integral as

$$G = \int_\infty^\infty e^{-\alpha x^2} dx = \sqrt{\frac{\pi}{\alpha}}$$ (1)

($\alpha$ is assumed constant) and take the dreivative of G with respect to $\alpha$. We get

$$\frac{\partial G}{\partial \alpha} = \int_\infty^\infty -x^2 ... ... dx = \sqrt{\pi} \frac{\partial\alpha^{-1/2}}{\partial \alpha}$$ (2)

which means

$$- \int_\infty^\infty x^2 e^{-\alpha x^2} dx = \sqrt{\pi} \left (-\frac{1}{2}\right ) \alpha^{-3/2}$$ (3)

so

$$\int_\infty^\infty x^2 e^{-\alpha x^2} dx = \sqrt{\pi} \frac{1}{2}\left (\frac{1}{ \alpha}\right )^{3/2}$$ (4)

which is a text-book, table-like result. We can repeat this again, and obtain

$$-\int_\infty^\infty x^4 e^{-\alpha x^2} dx = {\pi} \frac{\par... ...i}}{2}\left (\frac{1}{ \alpha}\right )^{3/2}}{\partial \alpha}$$ (5)

which would give

$$\int_\infty^\infty x^4 e^{-\alpha x^2} dx$$

and again, and again, so that finally we could see emerging

$$\int_\infty^\infty x^{2n} e^{-\alpha x^2} dx$$

where n is an integer. This is the form many tables use.