The Basic Solutions

The Hansen solutions to the vector HHE (that can expand the free space solutions for the vector potential or vector fields) are as follows. $\mbox{\boldmath$M$}_L$ is the (normalized) elementary solution consisting of a bessel function times $\mbox{\boldmath$L$}Y_L = \mbox{\boldmath$Y$}_{ll}^{m}$. It is (by construction) purely transverse: $\hat{\mbox{\boldmath$r$}}\cdot \mbox{\boldmath$M$}_L = 0$. $\mbox{\boldmath$N$}_L$ is the solution constructed by the taking the curl of $\mbox{\boldmath$M$}_L$. $\mbox{\boldmath$L$}_L$ is the ``longitudinal'' solution constructed by taking the gradient of the scalar solution - it is left as an exercise to show that this still satisfies the HHE. The three of these pieces span the range of possible solutions and reconstruct an identity tensor that can be used to construct a vector harmonic Green's function expansion.

This is summarized, with correction for factors of $k$ introduced by the derivatives, here:

$$\mbox{\boldmath$M$}_L = \frac{1}{\sqrt{\ell(\ell + 1)}} \mbox{\boldmath$L$}\left( f_\ell(... ...l(\ell + 1)}} f_\ell(kr)\mbox{\boldmath$Y$}_{ll}^{m}(\hat{\mbox{\boldmath$r$}})$$ (13.2)

$${\bf N}_L = \frac{i}{k} \mbox{\boldmath$\nabla$}\times {\bf M}_L$$ (13.3)

$${\bf L}_L = - \frac{i}{k} \mbox{\boldmath$\nabla$}\big( f_\ell(kr) Y_L(\hat{r}) \big)$$ (13.4)