Green's Functions for the Vector Helmholtz Equation

The correct form for the Green's function for the vector Helmholtz equation is

$$\stackrel{\Leftrightarrow}{\bf G}_\pm ({\bf r}, {\bf r}') = \stackrel{\Leftrightarrow}{\bf I} G_\pm({\bf r},{\bf r}')$$ (13.17)

(where $G_\pm({\bf r},{\bf r}')$ is a Green's function for the scalar IHE, that is:

$$G_\pm({\bf r},{\bf r}') = - \frac{e^{\pm i k R}}{4\pi R}$$ (13.18)

for $R = \mid {\bf r} - {\bf r}' \mid$. The identity tensor transforms a vector on the right into the same vector, so this seems like a trivial definition. However, the point is that we can now expand the identity tensor times the scalar Green's function in vector spherical harmonics or Hansen functions directly!

We get:

$$\stackrel{\Leftrightarrow}{\bf G}_\pm ({\bf r}, {\bf r}') = \mp i k \sum_{j,\ell,m} h_\ell^\pm(kr_>)j_\ell(kr_<) \mbox{\boldm... ...$}_{j \ell}^{m}(\hat{r}) \mbox{\boldmath$Y$}_{j \ell}^{m \ast}(\hat{r}') \hfill$$

$$= \mp i k \sum_{L} \bigg\{ {\bf M}_L^+({\bf r}_>) {\bf M}_L^{0 \ast} ({\bf r}_<) + {\bf N}_L^+({\bf r}_>) {\bf N}_L^{0 \ast}({\bf r}_<) +$$

$$\hfill {\bf L}_L^+({\bf r}_>) {\bf L}_L^{0 \ast}({\bf r}_<) \bigg\}$$ (13.19)

In all cases the ``*''s are to be considered sliding, able to apply to the $\mbox{\boldmath$Y$}_{jl}^{m}(\hat{\mbox{\boldmath$r$}})$ only of either term under an integral.

I do not intend to prove a key element of this assertion (that the products of the $\mbox{\boldmath$Y$}_{jl}^{m}(\hat{\mbox{\boldmath$r$}})$ involved reduce to Legendre polynomials in the angle between the arguments times the identity tensor) in class. Instead, I leave it as an exercise. To get you started, consider how similar completeness/addition theorems are proven for the spherical harmonics themselves from the given orthonormality relation.

With these relations in hand, we end our mathematical digression into vector spherical harmonics and the Hansen solutions and return to the land of multipolar radiation.