Inserting the plane-wave expansion into the brackets of equation (5) yields, $$ \begin{eqnarray*} \nonumber \int d{\hat{k}} Y_{lm}^*(\hat{k})\langle \mathbf{k}\vert \mathbf{r} \rangle & = & \left( {1\over 2\pi} \right) ^{3/2}4\pi i^{-l} j_l(kr) Y_{lm}^*(\hat{r}), \\ \nonumber \int d{\hat{k}'}\: Y_{lm}(\hat{k}') \langle \mathbf{r'}\vert \mathbf{k}' \rangle & = & \left( {1\over 2\pi} \right) ^{3/2}4\pi i^{l'} j_{l'}(k'r') Y_{l'm'}(\hat{r}). \end{eqnarray*} $$