linguistic:paraphrasing, stylistic:tone
Expansion in spherical harmonics

Most commonly, the series is written as a sum of spherical harmonics. Thus, we might write a function f (θ, φ) {\displaystyle f(\theta,\varphi)} as the sum f (θ, φ) = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ C ℓ m Y ℓ m (θ, φ) {\displaystyle f(\theta,\varphi)=\sum _{\ell =0}^{\infty }\,\sum _{m=-\ell }^{\ell }\,C_{\ell }^{m}\,Y_{\ell }^{m}(\theta,\varphi)} where Y ℓ m (θ, φ) {\displaystyle Y_{\ell }^{m}(\theta,\varphi)} are the standard spherical harmonics, and C ℓ m {\displaystyle C_{\ell }^{m}} are constant coefficients which depend on the function. The term C 0 0 {\displaystyle C_{0}^{0}} represents the monopole; C 1 − 1, C 1 0, C 1 1 {\displaystyle C_{1}^{-1},C_{1}^{0},C_{1}^{1}} represent the dipole; and so on. Equivalently, the series is also frequently written as f (θ, φ) = C + C i n i + C i j n i n j + C i j k n i n j n k + C i j k ℓ n i n j n k n ℓ + ⋯ {\displaystyle f(\theta,\varphi)=C+C_{i}n^{i}+C_{ij}n^{i}n^{j}+C_{ijk}n^{i}n^{j}n^{k}+C_{ijk\ell }n^{i}n^{j}n^{k}n^{\ell }+\cdots } where the n i {\displaystyle n^{i}} represent the components of a unit vector in the direction given by the angles θ {\displaystyle \theta } and φ {\displaystyle \varphi }, and indices are implicitly summed. Here, the term C {\displaystyle C} is the monopole; C i {\displaystyle C_{i}} is a set of three numbers representing the dipole; and so on.

In the above expansions, the coefficients may be real or complex. If the function being expressed as a multipole expansion is real, however, the coefficients must satisfy certain properties. In the spherical harmonic expansion, we must have C ℓ − m = (− 1) m C ℓ m ∗. {\displaystyle C_{\ell }^{-m}=(-1)^{m}C_{\ell }^{m\ast }\,.} In the multi-vector expansion, each coefficient must be real: C = C ∗; C i = C i ∗; C i j = C i j ∗; C i j k = C i j k ∗; … {\displaystyle C=C^{\ast };\ C_{i}=C_{i}^{\ast };\ C_{ij}=C_{ij}^{\ast };\ C_{ijk}=C_{ijk}^{\ast };\ \ldots }

While expansions of scalar functions are by far the most common application of multipole expansions, they may also be generalized to describe tensors of arbitrary rank. This finds use in multipole expansions of the vector potential in electromagnetism, or the metric perturbation in the description of gravitational waves.

For describing functions of three dimensions, away from the coordinate origin, the coefficients of the multipole expansion can be written as functions of the distance to the origin, r {\displaystyle r} —most frequently, as a Laurent series in powers of r {\displaystyle r}. For example, to describe the electromagnetic potential, V {\displaystyle V}, from a source in a small region near the origin, the coefficients may be written as: V (r, θ, φ) = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ C ℓ m (r) Y ℓ m (θ, φ) = ∑ j = 1 ∞ ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ D ℓ, j m r j Y ℓ m (θ, φ). {\displaystyle V(r,\theta,\varphi)=\sum _{\ell =0}^{\infty }\,\sum _{m=-\ell }^{\ell }C_{\ell }^{m}(r)\,Y_{\ell }^{m}(\theta,\varphi)=\sum _{j=1}^{\infty }\,\sum _{\ell =0}^{\infty }\,\sum _{m=-\ell }^{\ell }{\frac {D_{\ell,j}^{m}}{r^{j}}}\,Y_{\ell }^{m}(\theta,\varphi).}