Hankel transform (${\mathcal{H}}_{0}f(\nu )={\int }_0^{\infty }f(r)J_0(\nu r)rdr$)

The order-$\nu$ Hankel transform replaces the exponential kernel of a Fourier transform by a Bessel function:Under suitable integrability and inversion hypotheses, $f(r)={\int }_0^{\infty }({\mathcal{H}}_{\nu }f)(k)J_{\nu }(kr)kdk$. The measures $rdr$ and $kdk$ belong to this convention and cannot be discarded.

For a radial function $F(x)=f(|x|)$ on the plane, angular integration gives $\hat{F}(\xi )=2\pi ({\mathcal{H}}_{0}f)(|\xi |)$ when the Fourier transform kernel is $e^{-ix\cdot \xi }$. Thus rotational symmetry reduces a two-dimensional transform to a one-dimensional radial integral.

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 $\Phi /(2\pi )$.

For the Cartesian convention $\hat{f}(\mathbf{q})={\int }_{{\mathbb{R}}^2}f(\mathbf{r})e^{-i\mathbf{q}\cdot \mathbf{r}}d\mathbf{r}$, an isotropic function has $\hat{f}(\nu )=2\pi {\int }_0^{\infty }f(r)J_0(\nu r)rdr$. Its inverse is $f(r)=(2\pi {)}^{-1}{\int }_0^{\infty }\hat{f}(\nu )J_0(\nu r)\nu d\nu$. The self-reciprocal Hankel transform instead uses $F_H=\hat{f}/(2\pi )$. In particular $f(0)=\int F_H(\nu )\nu d\nu$. Mixing these spectral conventions changes attenuation coefficients by $2\pi$.