Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 69 / 1 / c / Solution

Write the transverse displacement as $\mathbf{s}$ and take the common line-of-sight interval to be $[0,L]$. The rotation measure is a line integral of the magnetic field, so its correlation isSubstitute the Fourier inversion of the solenoidal isotropic spectral tensor. Keeping the finite observation length gives the exact windowed expressionThe allowed extension of the longitudinal displacement integral to the whole real line is the long-path projection of a magnetic correlation. It applies when $L$ is much larger than the correlation length: equivalently, the squared window becomes $2\pi L\delta (k_z)$ inside the integral. The Dirac delta function sets $k_z=0$, leavingStatistical isotropy makes this a function of $s=|\mathbf{s}|$. The displayed two-dimensional relation is the intended long-path approximation; the preceding windowed formula accounts for finite-length edge effects.