Spherical harmonics and partial-wave conventions in three dimensions

Assume Countable Choice and Dependent Choice. Use the unit sphere $S^2$ and its polar surface measure $d\sigma$ from The polar surface set function on the unit sphere. For each integer $\mathrm{\ell }\ge 0$, let ${\mathcal{H}}_{\mathrm{\ell }}$ be the complex vector space of restrictions to $S^2$ of homogeneous degree-$\mathrm{\ell }$ polynomials $P$ on ${\mathbb{R}}^3$ satisfying the Euclidean harmonic equation $\mathrm{\Delta }P=0$. For complex-valued polynomials or functions, $\mathrm{\Delta }$ and its polar-coordinate formulas act componentwise on real and imaginary parts. In polar coordinates $x=r\omega$, define ${\mathrm{\Delta }}_{S^2}$ by ${\mathrm{\Delta }}_{{\mathbb{R}}^3}={\mathrm{\partial }}_r^2+\frac{2}{r}{\mathrm{\partial }}_r+\frac{1}{r^2}{\mathrm{\Delta }}_{S^2},$ equivalently ${\mathrm{\Delta }}_{S^2}=\frac{1}{\sin \theta }{\mathrm{\partial }}_{\theta }(\sin \theta {\mathrm{\partial }}_{\theta })+\frac{1}{{\sin }^2\theta }{\mathrm{\partial }}_{\varphi }^2$ away from the poles. Thus if $Y\in {\mathcal{H}}_{\mathrm{\ell }}$ is the restriction of $P$, then ${\mathrm{\Delta }}_{S^2}Y=-\mathrm{\ell }(\mathrm{\ell }+1)Y$.

For $\mathrm{\ell }\ge 0$, define the normalized zonal polynomial $P_{\mathrm{\ell }}(t)$ to be the degree-$\mathrm{\ell }$ polynomial satisfying $(1-t^2)P_{\mathrm{\ell }}^{\mathrm{\prime }\mathrm{\prime }}(t)-2t{P}_{\mathrm{\ell }}^{\mathrm{\prime }}(t)+\mathrm{\ell }(\mathrm{\ell }+1)P_{\mathrm{\ell }}(t)=0,P_{\mathrm{\ell }}(1)=1.$ The later harmonic-basis lemma proves existence and uniqueness of this normalized polynomial and identifies it with the usual Legendre convention.

The term partial wave of degree $\mathrm{\ell }$ means the ${\mathcal{H}}_{\mathrm{\ell }}$ component of an angular expansion in $L^2(S^2,d\sigma ;\mathbb{C})$. Completeness, orthogonality, finite dimension, and the evaluation bound for these components are established in Spherical harmonics form a complete basis with the addition formula ↗.

For a radial field $u(r,\omega )$, its degree-$\mathrm{\ell }$ component is written $u_{\mathrm{\ell }}(r,\omega )\in {\mathcal{H}}_{\mathrm{\ell }}$; any orthonormal basis within this finite-dimensional space gives the equivalent scalar mode equations. The definition does not select such a basis globally.