Isotropic transverse covariance spectrum ($C(\rho )=\frac{1}{2\pi }{\int }_0^{\infty }\Phi (\kappa )J_0(\kappa \rho )\kappa d\kappa$)

For the Fourier transform convention $C(\mathbf{s})=\int \Phi (\mathbf{\kappa })e^{i\mathbf{\kappa }\cdot \mathbf{s}}{d}^2\kappa /(2\pi {)}^2$, angular integration of an isotropic power spectrum gives the displayed Hankel transform. A purely radial Hankel transform without the prefactor corresponds to defining its spectrum as $\Phi /(2\pi )$.