# Connection between $\nu=\frac{N}{2}-1^\text{th}$ order Hankel transforms and hyperspherically symmetric functions?

In The Transforms And Applications Handbook 2nd edition chapter 9 (Hankel Transforms), Piessens briefly mentions that the Fourier transform of an $N$-dimensional hyperspherically symmetric function has a radial profile given by the Hankel transform of order $\nu=\frac{N}{2}-1$ of the input radial profile.

However, he does not include any citation, and does not prove the result (the chapter is mainly devoted to the case $N=2$). I believe it is wrong, because in the case $N=3$, we have $$\mathcal{F}\left(f(|\mathbf{r}|),\mathbf{r},\mathbf{k}\right)=\sqrt{\frac{2}{\pi}}\int_0^\infty f(r)r^2 j_0(kr)\,\mathrm{d}r\\=\sqrt{\frac{2}{\pi}}\int_0^\infty f(r)r\frac{\sin(kr)}{k}\,\mathrm{d}r$$ where $j_0$ is the zeroth order spherical Bessel function, whereas $$\mathcal{H}_{1/2}\left(f(|\mathbf{r}|),\mathbf{r},\mathbf{k}\right)=\int_0^\infty f(r)rJ_{1/2}(kr)\,\mathrm{d}r=\sqrt{\frac{2}{\pi}}\int_0^\infty f(r)\sqrt{r}\frac{\sin(kr)}{\sqrt{k}}\,\mathrm{d}r$$ which are not the same.

Is this an error? If so, what is the correct $N$-dimensional analogue of the Fourier transform of radially-symmetric functions?

$$\mathcal{F}\left(f(|\mathbf{r}|),\mathbf{r},\mathbf{k}\right)= \frac{2\pi}{k^{N/2-1}}\mathcal{H}_{N/2-1}(f(r)r^{N/2-1},r,2\pi k).$$