 Steady toroidal induction by spherical differential rotation

An axial uniform magnetic field and localized radial differential rotation generate a steady toroidal magnetic field. The angular factor $\sin \theta \cos \theta$ is an eigenfunction of the toroidal angular diffusion operator with eigenvalue $-6$. Its radial amplitude satisfies $\eta (f^{\prime \prime }+2f^{\prime }/r-6f/r^2)=-B_{0}r{\Omega }^{\prime }$. Differentiating the integral in the title proves the formula. It is regular at the origin and decays at infinity when $\Omega$ is finite at zero and decays sufficiently rapidly.