Editing Multipole expansion (section)

===Spherical form===
The potential {{math|''V''('''R''')}} at a point {{math|'''R'''}} outside the charge distribution, i.e. {{math|{{abs|'''R'''}} > ''r''<sub>max</sub>}}, can be expanded by the [[Laplace expansion (potential)|Laplace expansion]]:
<math display="block">V(\mathbf{R}) \equiv \sum_{i=1}^N \frac{q_i}{4\pi \varepsilon_0 |\mathbf{r}_i - \mathbf{R}|}
=\frac{1}{4\pi \varepsilon_0} \sum_{\ell=0}^\infty \sum_{m=-\ell}^{\ell}
(-1)^m I^{-m}_\ell(\mathbf{R}) \sum_{i=1}^N q_i R^m_\ell(\mathbf{r}_i),</math>
where <math>I^{-m}_{\ell}(\mathbf{R})</math> is an irregular [[solid harmonic]] (defined below as a [[spherical harmonic]] function divided by <math>R^{\ell+1}</math>) and <math>R^m_{\ell}(\mathbf{r})</math> is a regular solid harmonic (a spherical harmonic times {{math|r<sup>''ℓ''</sup>}}). We define the ''spherical multipole moment'' of the charge distribution as follows
<math display="block">Q^m_\ell \equiv \sum_{i=1}^N q_i R^m_\ell(\mathbf{r}_i),\quad\ -\ell \le m \le \ell.</math>
Note that a multipole moment is solely determined by the charge distribution (the positions and magnitudes of the ''N'' charges).

A [[spherical harmonic]] depends on the unit vector <math>\hat{R}</math>. (A unit vector is determined by two spherical polar angles.) Thus, by definition, the irregular solid harmonics can be written as
<math display="block">I^m_{\ell}(\mathbf{R}) \equiv \sqrt{\frac{4\pi}{2\ell+1}} \frac{Y^m_{\ell}(\hat{R})}{R^{\ell+1}}</math>
so that the ''multipole expansion'' of the field {{math|''V''('''R''')}} at the point {{math|'''R'''}} outside the charge distribution is given by

<math display="block">\begin{align}
V(\mathbf{R})
& = \frac{1}{4\pi\varepsilon_{0}}\sum_{\ell=0}^{\infty}
\sum_{m=-\ell}^{\ell}(-1)^{m} I^{-m}_{\ell}(\mathbf{R}) Q^{m}_{\ell}\\
& = \frac{1}{4\pi\varepsilon_{0}}\sum_{\ell=0}^{\infty}\left[\frac{4\pi}{2\ell + 1}\right]^{1/2}\;\frac{1}{R^{\ell + 1}}
\sum_{m=-\ell}^{\ell}(-1)^{m} Y^{-m}_{\ell}(\hat{R}) Q^{m}_{\ell}, \qquad R > r_{\mathrm{max}}
\end{align}</math>

This expansion is completely general in that it gives a closed form for all terms, not just for the first few. It shows that the [[spherical multipole moments]] appear as coefficients in the {{math|1/''R''}} expansion of the potential.

It is of interest to consider the first few terms in real form, which are the only terms commonly found in undergraduate textbooks.
Since the summand of the ''m'' summation is invariant under a unitary transformation of both factors simultaneously and since transformation of complex spherical harmonics to real form is by a [[Solid harmonics#Real form|unitary transformation]], we can simply substitute real irregular solid harmonics and real multipole moments. The {{math|1=''ℓ'' =&thinsp;0}} term becomes
<math display="block">V_{\ell=0}(\mathbf{R}) =
\frac{q_\mathrm{tot}}{4\pi \varepsilon_0 R} \quad\hbox{with}\quad q_\mathrm{tot}\equiv\sum_{i=1}^N q_i.</math>
This is in fact [[Coulomb's law]] again. For the {{math|1=''ℓ'' =&thinsp;1}} term we introduce
<math display="block">\mathbf{R} = (R_x, R_y, R_z),\quad \mathbf{P} = (P_x, P_y, P_z)\quad
\hbox{with}\quad P_\alpha \equiv \sum_{i=1}^N q_i r_{i\alpha}, \quad \alpha=x,y,z.</math>
Then
<math display="block">V_{\ell=1}(\mathbf{R}) =
\frac{1}{4\pi \varepsilon_0 R^3} (R_x P_x +R_y P_y + R_z P_z) = \frac{\mathbf{R} \cdot \mathbf{P} }{4\pi \varepsilon_0 R^3} =
\frac{\hat\mathbf{R} \cdot \mathbf{P} }{4\pi \varepsilon_0 R^2}.</math>
This term is identical to the one found in Cartesian form.

In order to write the {{math|1=''ℓ'' =&thinsp;2}} term, we have to introduce shorthand notations for the five real components of the quadrupole moment and the real spherical harmonics. Notations of the type
<math display="block">Q_{z^2} \equiv \sum_{i=1}^N q_i\; \frac{1}{2}(3z_i^2 - r_i^2),</math>
can be found in the literature. Clearly the real notation becomes awkward very soon, exhibiting the usefulness of the complex notation.