Editing Spherical harmonics (section)

==Harmonic polynomial representation==
{{see also|#Higher dimensions}}

The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions <math>\R^3 \to \Complex</math>. Specifically, we say that a (complex-valued) polynomial function <math>p: \R^3 \to \Complex</math> is [[Homogeneous polynomial|''homogeneous'']] of degree <math>\ell</math> if
<math display="block">p(\lambda\mathbf x)=\lambda^\ell p(\mathbf x)</math>
for all real numbers <math>\lambda \in \R</math> and all <math>\mathbf x \in \R^3</math>. We say that <math>p</math> is [[Harmonic function|''harmonic'']] if
<math display="block">\Delta p=0,</math>
where <math>\Delta</math> is the [[Laplacian]]. Then for each <math>\ell</math>, we define
<math display="block">\mathbf{A}_\ell = \left\{\text{harmonic polynomials } \R^3 \to \Complex \text{ that are homogeneous of degree } \ell \right\}.</math>

For example, when <math>\ell=1</math>, <math>\mathbf{A}_1</math> is just the 3-dimensional space of all linear functions <math>\R^3 \to \Complex</math>, since any such function is automatically harmonic. Meanwhile, when <math>\ell = 2</math>, we have a 6-dimensional space:
<math display="block">\mathbf{A}_2 = \operatorname{span}_{\Complex}(x_1 x_2,\, x_1 x_3,\, x_2 x_3,\, x_1^2,\, x_2^2, \, x_3^2).</math>

A general formula for the dimension, <math>d_l</math>, of the set of homogenous polynomials of degree  <math>\ell</math> in <math>\R^n</math> is<ref>{{harvnb|Stein|Weiss|1971|p=139}}</ref>
<math display="block">d_l = \frac{(n + l - 1)!}{(n-1)! \, l!}</math>
For any <math>\ell</math>, the space <math>\mathbf{H}_{\ell}</math> of spherical harmonics of degree <math>\ell</math> is just the space of restrictions to the sphere <math>S^2</math> of the elements of <math>\mathbf{A}_\ell</math>.<ref>{{harvnb|Hall|2013}} Section 17.6</ref> As suggested in the introduction, this perspective is presumably the origin of the term “spherical harmonic” (i.e., the restriction to the sphere of a [[harmonic function]]).

For example, for any <math>c \in \Complex</math> the formula
<math display="block">p(x_1, x_2, x_3) = c(x_1 + ix_2)^\ell</math>
defines a homogeneous polynomial of degree <math>\ell</math> with domain and codomain <math>\R^3 \to \Complex</math>, which happens to be independent of <math>x_3</math>. This polynomial is easily seen to be harmonic. If we write <math>p</math> in spherical coordinates <math>(r,\theta,\varphi)</math> and then restrict to <math>r = 1</math>, we obtain
<math display="block">p(\theta,\varphi) = c \sin(\theta)^\ell (\cos(\varphi) + i \sin(\varphi))^\ell,</math>
which can be rewritten as
<math display="block">p(\theta,\varphi) = c\left(\sqrt{1-\cos^2(\theta)}\right)^\ell e^{i\ell\varphi}.</math>
After using the formula for the [[Associated Legendre polynomials|associated Legendre polynomial]] <math>P^\ell_\ell</math>, we may recognize this as the formula for the spherical harmonic <math>Y^\ell_\ell(\theta, \varphi).</math><ref>{{harvnb|Hall|2013}} Lemma 17.16</ref> (See [[#Special cases and values|Special cases]].)