Use the operator separation technique to show that
(a) the GF for the IIelmholtz operator \(\nabla^{2}+k^{2}\) in three dimensions
is
$$
G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=-i k \sum_{l=0}^{\infty}
\sum_{m=-l}^{l} j_{l}\left(k r_{<}\right) h_{l}\left(k r_{>}\right) Y_{l
m}(\theta, \varphi) Y_{l m}^{*}\left(\theta^{\prime}, \varphi^{\prime}\right),
$$
where \(r_{<}\left(r_{>}\right)\) is the smaller (larger) of \(r\) and
\(r^{\prime}\) and \(j_{l}\) and \(h_{l}\) are the spherical Bessel and Hankel
functions, respectively. No explicit BCs are assumed except that there is
regularity at \(r=0\) and that \(G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)
\rightarrow\) 0 for \(|\mathbf{r}| \rightarrow \infty\)
(b) Obtain the identity
$$
\frac{e^{i k\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}}{4
\pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}=i k \sum_{l=0}^{\infty}
\sum_{m=-l}^{l} j_{l}\left(k r_{<}\right) h_{l}\left(k r_{>}\right) Y_{l
m}(\theta, \varphi) Y_{l m}^{*}\left(\theta^{\prime}, \varphi^{\prime}\right)
.
$$
(c) Derive the plane wave expansion [see Eq. (19.46)]
$$
e^{i \mathbf{k} \cdot \mathbf{r}}=4 \pi \sum_{l=0}^{\infty} \sum_{m=-l}^{l}
i^{l} j_{l}(k r) Y_{l m}^{*}\left(\theta^{\prime}, \varphi^{\prime}\right)
Y_{l m}(\theta, \varphi),
$$
where \(\theta^{\prime}\) and \(\varphi^{\prime}\) are assumed to be the angular
coordinates of \(\mathbf{k}\). Hint:
Let \(\left|\mathbf{r}^{\prime}\right| \rightarrow \infty\), and use
$$
\left|\mathbf{r}-\mathbf{r}^{\prime}\right|=\left(r^{\prime 2}+r^{2}-2
\mathbf{r} \cdot \mathbf{r}^{\prime}\right)^{1 / 2} \rightarrow
r^{\prime}-\frac{\mathbf{r}^{\prime} \cdot \mathbf{r}}{r^{\prime}}
$$
and the asymptotic formula \(h_{l}^{(1)}(z) \rightarrow(1 / z) e^{i[z+(l+1)(\pi
/ 2)]}\), valid for large \(z\).
