= Hankel inversion of a rotation-measure correlation
{c}
{title2=$H(k)=\frac{2\pi}{a_0^2n_e^2L}\int_0^\infty rJ_0(kr)C_{\mathrm{RM}}(r)\,dr$}
Invert the two-dimensional <Fourier transform> in the <long-path projection of a magnetic correlation>, then perform the angular integration in <polar coordinates>. The angular integral is $2\pi J_0(kr)$, a <Bessel function of the first kind>, giving the displayed <Hankel transform> with the stated <Fourier-Hankel normalization for an isotropic spectrum>. The result recovers the scalar tensor coefficient $H$, whose shell-integrated <magnetic energy spectrum> has an additional factor $k^2$.
Back to article page
