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I've seen in Jackson's 4.9 problem and also in a similar example given in the book (pag. 157) they use the azimuthal symmetry solution for Laplace equation which gives the ordinary Legendre polynomials. The 4.9 problem states:

Find a potential at all points in space for a point charge $q$ located in a free space a distance $d$ from the center of a solid dielectric sphere of radius $a $ $(a<d)$ and dielectric constant $ϵ/ϵ_0$.

And part of the solution is to find the potential inside and outside the sphere:

$$\Phi_{\mathrm{in}}(\vec{r})=\sum_{l=0}^{\infty} A_{l} r^{l} P_{l}(\cos \theta)$$ $$\Phi_{\text {out }}(\vec{r})=\Phi_{q}+\sum_{l=0}^{\infty} B_{l} r^{-(l+1)} P_{l}(\cos \theta)$$

where $\Phi_{q}$ is the potential produced by the point charge.

Why the azimuthal symmetry is assumed? Why not working with the general solution involving spherical harmonics?

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    $\begingroup$ If the point charge is located on the $z$ axis you will have symmetry in $\phi$. Do you agree? $\endgroup$
    – Newbie
    Jan 20 at 22:46

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