= Fourier-Hankel normalization for an isotropic spectrum
{c}
{title2=$\widehat f(\nu)=2\pi\mathcal H_0f(\nu)$}
For the Cartesian convention $\widehat f(\mathbf q)=\int_{\mathbb R^2}f(\mathbf r)e^{-i\mathbf q\cdot\mathbf r}d\mathbf r$, an isotropic function has $\widehat f(\nu)=2\pi\int_0^\infty f(r)J_0(\nu r)r\,dr$. Its inverse is $f(r)=(2\pi)^{-1}\int_0^\infty\widehat f(\nu)J_0(\nu r)\nu\,d\nu$. The self-reciprocal <Hankel transform> instead uses $F_H=\widehat f/(2\pi)$. In particular $f(0)=\int F_H(\nu)\nu\,d\nu$. Mixing these spectral conventions changes attenuation coefficients by $2\pi$.
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