Integral on $\mathbb{R}^d$

Question

This is probably a simple question, but I don't have tons of experience integrating on $\mathbb{R}^d$ for arbitrary $d$. I'd like to compute the following integral $$ \int_{\mathbb{R}^d} e^{-|\eta|^2/2} |\eta|^{\alpha-d}d\eta $$ where $|\cdot|$ is the standard Euclidean norm. Is this something that requires the use of a gamma function identity? Or, simply integration by parts a number of times? I'd appreciate any help you can give me.

Answer 1

I'm assuming $\alpha > 0$. The Gamma function will be involved. Since the integrand is a radial function, it will be useful to switch to spherical coordinates first before computing the integral.

\begin{align}\int_{\Bbb R^d} e^{-|\eta|^2}|\eta|^{\alpha - d}\, d\eta &= \int_0^\infty \int_{S^{d-1}} e^{-r^2/2} r^{\alpha - d} r^{d - 1}\, d\omega\, dr \\ &= |S^{d-1}| \int_0^\infty e^{-r^2/2} r^{\alpha - 1}\, dr\\ &= |S^{d-1}|\int_0^\infty e^{-u} (2u)^{\frac{\alpha - 2}{2}}\, du & (\text{letting $u = r^2/2$})\\ &= 2^{\frac{\alpha-2}{2}}|S^{d-1}|\Gamma\left(\frac{\alpha}{2}\right)\\ &= 2^{\alpha/2}\pi^{d/2}\frac{\Gamma\left(\frac{\alpha}{2}\right)}{\Gamma\left(\frac{d}{2}\right)} \end{align}

The last equality follows from the formula $|S^{d-1}| = 2\pi^{d/2}/\Gamma(d/2)$.