Theorem 0.1.1 The multipole expansion theorem states that for a charge distribution localized within a sphere, the electric potential outside the sphere can be written in terms of spherical harmonics.
\[V(\vec{r}) = \frac{1}{4\pi\varepsilon_0}\sum_{\ell=0}^\infty\sum_{m=-\ell}^\ell\frac{4\pi}{2\ell+1}q_{\ell,m}\frac{Y_{\ell,m}(\theta,\phi)}{r^{\ell+1}}\]\[q_{\ell,m} = \int Y^*_{\ell,m}(\theta,\phi)r^{\ell}\rho(\vec{r})d^3r\]
Result 0.1.2 The multipole moment $q_{\ell,m}$ and $q_{\ell, -m}$ are related by the relation $q_{\ell,-m} = (-1)^{m}q^*_{\ell,m}$.
Definition 0.1.3 The monopole moment $q$ of a charge distribution $\rho(\vec{r})$ is simply the total charge.
\[q = \int \rho(\vec{r}) d^3r\]
Definition 0.1.4 The dipole moment $\vec{p}$ of a charge distribution $\rho(\vec{r})$ is defined with the following integral.
\[\vec{p} = \int\vec{r}\rho(\vec{r})d^3r\]
Definition 0.1.5 The quadrupole moment $Q$ of a charge distribution $\rho(\vec{r})$ is a $3$ by $3$ matrix with components $Q_{j,k}$ defined with the following integral.
\[Q_{j,k} = \int \left[3r_jr_k-r^2\delta_{j,k}\right]\rho(\vec{r})d^3r\]
Result 0.1.6 The multipole expansion of a charge distribution can be written in terms of the multiple moments of the charge distribution. The first 6 terms of the multiple expansion are written below in terms of the monopole moment $q$, the dipole moment $\vec{p}$ and the quadrupole moment $Q$ of a charge distribution $\rho(\vec{r})$.
\[q_{0,0} = \frac{1}{\sqrt{4\pi}}\int\rho(\vec{r})d^3r = \frac{1}{\sqrt{4\pi}}q\]\[q_{1,1} = -\sqrt{\frac{3}{8\pi}}\int(x-iy)\rho(\vec{r})d^3r = -\sqrt{\frac{3}{8\pi}}(p_x-ip_y)\]\[q_{1,0} = \sqrt{\frac{3}{4\pi}}\int z\rho(\vec{r})d^3r = \sqrt{\frac{3}{4\pi}}p_z\]\[q_{2,2} = \frac{1}{4}\sqrt{\frac{15}{2\pi}}\int(x-iy)\rho(\vec{r})d^3r = \frac{1}{12}\sqrt{\frac{15}{2\pi}}(Q_{1,1}-2iQ_{1,2}-Q_{2,2})\]\[q_{2,1} = -\sqrt{\frac{15}{8\pi}}\int z(x-iy)\rho(\vec{r})d^3r = -\frac{1}{3}\sqrt{\frac{15}{8\pi}}(Q_{1,3}-iQ_{2,3})\]\[q_{2,0} = \frac{1}{2}\sqrt{\frac{5}{4\pi}}\int(3z^2-r^2)\rho(\vec{r})d^3r = \frac{1}{2}\sqrt{\frac{5}{4\pi}}Q_{3,3}\]