Editing Gaussian integral (section)

==Relation to the gamma function==

The integrand is an [[even function]],

<math display="block">\int_{-\infty}^{\infty} e^{-x^2} dx = 2 \int_0^\infty e^{-x^2} dx</math>

Thus, after the change of variable <math display="inline">x = \sqrt{t}</math>, this turns into the Euler integral

<math display="block">2 \int_0^\infty e^{-x^2} dx = 2\int_0^\infty \frac{1}{2}\ e^{-t} \ t^{-\frac{1}{2}} dt = \Gamma{\left(\frac{1}{2}\right)} = \sqrt{\pi}</math>

where <math display="inline"> \Gamma(z) = \int_{0}^{\infty} t^{z-1} e^{-t} dt </math> is the [[gamma function]]. This shows why the [[factorial]] of a half-integer is a rational multiple of <math display="inline">\sqrt \pi</math>. More generally,
<math display="block">\int_0^\infty x^n e^{-ax^b} dx = \frac{\Gamma{\left((n+1)/b\right)}}{b a^{(n+1)/b}}, </math>
which can be obtained by substituting <math>t=a x^b</math> in the integrand of the gamma function  to get <math display="inline"> \Gamma(z) = a^z b \int_0^{\infty} x^{bz-1} e^{-a x^b} dx </math>.