By transforming to spherical polar coordinates in (mathbb{R}^{d}), show that [ int_{mathbb{R}^{d}} frac{d omega}{left(1+|omega|^{2}ight)^{v+d / 2}}=A_{d-1} int_{0}^{infty}

Question:

By transforming to spherical polar coordinates in \(\mathbb{R}^{d}\), show that

\[
\int_{\mathbb{R}^{d}} \frac{d \omega}{\left(1+\|\omega\|^{2}ight)^{v+d / 2}}=A_{d-1} \int_{0}^{\infty} \frac{x^{d-1} d x}{\left(1+x^{2}ight)^{v+d / 2}}
\]

where \(A_{d-1}=2 \pi^{d / 2} / \Gamma(d / 2)\) is the surface area of the unit sphere in \(\mathbb{R}^{d}\). For \(v>-d / 2\), deduce that the Matérn measure on \(\mathbb{R}^{d}\)

\[
M_{d}(d \omega)=\frac{\Gamma(u+d / 2) d \omega}{\pi^{d / 2}\left(1+\|\omega\|^{2}ight)^{v+d / 2}} .
\]

has finite mass if and only if \(v>0\). Show that the total mass is a constant independent of the dimension of the space.