TITLE: Why does Jackson's book use different method to calculate the vectorial multiple expansion coefficients in chapter 10.3 and 9.7?
QUESTION [0 upvotes]: In chapter 9.7:
$$Z_0a_E(l,m)f_l(kr)=-\frac{k}{\sqrt{l(l+1)}}\int{Y^*_{lm}\mathbf{r\cdot E}d\Omega}\tag{9.123}$$
In chapter 10.3
$$a_\pm(l,m)j_l(kr)=\int{\mathbf X^*_{lm}\mathbf\cdot\mathbf E(\mathbf x)d\Omega}\tag{10.49}$$
10.3 should be a special case of the theory in 9.7, then 9.123 should be able to be used in place of 10.49 (even it may be not as straight forward as using 10.49), is that true?
What is the factor that let chapter 10.3 to use a different set of orthogonality properties (i.e. 10.48) and 10.49 instead of 9.123?

REPLY [0 votes]: $$\mathbf E(\mathbf x)=(\boldsymbol{\epsilon}_1\pm i \boldsymbol{\epsilon}_2)e^{ikz}\tag{10.46}$$
With
$$a_\pm(l,m)j_l(kr)=\int{\mathbf X^*_{lm}\mathbf\cdot\mathbf E(\mathbf x)d\Omega}\tag{10.49}$$
after the  expansion of the dot product, it finally becomes:
$$a_\pm(l,m)j_l(kr)=\frac{\sqrt{(l\pm m)(l\mp m+1)}}{\sqrt{l(l+1)}}\int{Y^*_{l,m\mp 1}e^{ikz}d\Omega}\tag{10.52}$$
and $e^{ikz}$ has a simple spherical harmonic expansion since:
$$e^{i\mathbf {k\cdot x}}=\sum_{l=0}^{\infty}i^l\sqrt{4\pi (2l+1)}j_l(kr)Y_{l,0}(\gamma) \tag{10.45}$$
where $\gamma$ is the angle between $\mathbf k$ and $\mathbf x$ which should be zero for the expansion of $e^{ikz}$.
This expansion of $e^{ikz}$ bring 10.52 to a very simple result (i.e. orthogonality of the spherical harmonics)
On the other hand if
$$Z_0a_E(l,m)f_l(kr)=-\frac{k}{\sqrt{l(l+1)}}\int{Y^*_{lm}\mathbf{r\cdot E}d\Omega}\tag{9.123}$$
is used instaed, since $\mathbf {r\cdot E}$ ($\mathbf E$ from 10.46 above)doesn't have a easy expansion like 10.45, the orthogonality can't be used to get a simple result form.
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