Editing Laplace's equation (section)

=== Laplace's spherical harmonics ===
{{Main|Spherical harmonics#Laplace's spherical harmonics}}
[[File:Rotating_spherical_harmonics.gif|right|thumb|Real (Laplace) spherical harmonics {{math|''Y<sub>ℓ</sub><sup>m</sup>''}} for {{math|1=''ℓ'' = 0, ..., 4}} (top to bottom) and {{math|1=''m'' = 0, ..., ''ℓ''}} (left to right). Zonal, sectoral, and tesseral harmonics are depicted along the left-most column, the main diagonal, and elsewhere, respectively. (The negative order harmonics <math>Y_{\ell}^{-m}</math> would be shown rotated about the {{math|''z''}} axis by <math>90^\circ/m</math> with respect to the positive order ones.)]]
Laplace's equation in [[Spherical coordinate system|spherical coordinates]] is:<ref>The approach to spherical harmonics taken here is found in {{harv|Courant|Hilbert|1962|loc=§V.8, §VII.5}}.</ref>

<math display="block"> \nabla^2 f = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial f}{\partial r}\right) 
 + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta}\left(\sin\theta \frac{\partial f}{\partial \theta}\right) 
 + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2 f}{\partial \varphi^2} = 0.</math>

Consider the problem of finding solutions of the form {{math|1=''f''(''r'', ''θ'', ''φ'') = ''R''(''r'') ''Y''(''θ'', ''φ'')}}.  By [[Separation of variables#pde|separation of variables]], two differential equations result by imposing Laplace's equation:

<math display="block">\frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) = \lambda,\qquad \frac{1}{Y}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta \frac{\partial Y}{\partial\theta}\right) + \frac{1}{Y}\frac{1}{\sin^2\theta}\frac{\partial^2Y}{\partial\varphi^2} = -\lambda.</math>

The second equation can be simplified under the assumption that {{math|''Y''}} has the form {{math|1=''Y''(''θ'', ''φ'') = Θ(''θ'') Φ(''φ'')}}. Applying separation of variables again to the second equation gives way to the pair of differential equations

<math display="block">\frac{1}{\Phi} \frac{d^2 \Phi}{d\varphi^2} = -m^2</math>
<math display="block">\lambda\sin^2\theta + \frac{\sin\theta}{\Theta} \frac{d}{d\theta} \left(\sin\theta \frac{d\Theta}{d\theta}\right) = m^2</math>

for some number {{math|''m''}}. A priori, {{math|''m''}} is a complex constant, but because {{math|Φ}} must be a [[periodic function]] whose period evenly divides {{math|2''π''}}, {{math|''m''}} is necessarily an integer and {{math|Φ}} is a linear combination of the complex exponentials {{math|''e''<sup>±''imφ''</sup>}}. The solution function {{math|''Y''(''θ'', ''φ'')}} is regular at the poles of the sphere, where {{math|1=''θ'' = 0, ''π''}}.  Imposing this regularity in the solution {{math|Θ}} of the second equation at the boundary points of the domain is a [[Sturm–Liouville problem]] that forces the parameter {{math|''λ''}} to be of the form {{math|1=''λ'' = ''ℓ'' (''ℓ'' + 1)}} for some non-negative integer with {{math|''ℓ'' ≥ {{!}}''m''{{!}}}}; this is also explained [[Spherical harmonics#Orbital angular momentum|below]] in terms of the [[Angular momentum operator|orbital angular momentum]].  Furthermore, a change of variables {{math|1=''t'' = cos ''θ''}} transforms this equation into the [[Associated Legendre function|Legendre equation]], whose solution is a multiple of the [[associated Legendre polynomial]] {{math|''P<sub>ℓ</sub><sup>m</sup>''(cos ''θ'')}} .  Finally, the equation for {{math|''R''}} has solutions of the form {{math|1=''R''(''r'') = ''A r<sup>ℓ</sup>'' + ''B r''<sup>−''ℓ'' − 1</sup>}}; requiring the solution to be regular throughout {{math|'''R'''<sup>3</sup>}} forces {{math|1=''B'' = 0}}.<ref group=note>Physical applications often take the solution that vanishes at infinity, making {{math|1=''A'' = 0}}.  This does not affect the angular portion of the spherical harmonics.</ref>

Here the solution was assumed to have the special form {{math|1=''Y''(''θ'', ''φ'') = Θ(''θ'') Φ(''φ'')}}.  For a given value of {{math|''ℓ''}}, there are {{math|2''ℓ'' + 1}} independent solutions of this form, one for each integer {{math|''m''}} with {{math|−''ℓ'' ≤ ''m'' ≤ ''ℓ''}}.  These angular solutions are a product of [[trigonometric function]]s, here represented as a [[Euler's formula|complex exponential]], and associated Legendre polynomials:
<math display="block"> Y_\ell^m (\theta, \varphi ) = N e^{i m \varphi } P_\ell^m (\cos{\theta} )</math>
which fulfill
<math display="block"> r^2\nabla^2 Y_\ell^m (\theta, \varphi ) = -\ell (\ell + 1 ) Y_\ell^m (\theta, \varphi ).</math>

Here {{math|''Y<sub>ℓ</sub><sup>m</sup>''}} is called a spherical harmonic function of degree {{mvar|ℓ}} and order {{mvar|m}}, {{math|''P<sub>ℓ</sub><sup>m</sup>''}} is an [[associated Legendre polynomial]], {{math|''N''}} is a normalization constant, and {{mvar|θ}} and {{mvar|φ}} represent colatitude and longitude, respectively. In particular, the [[colatitude]] {{mvar|θ}}, or polar angle, ranges from {{math|0}} at the North Pole, to {{math|''π''/2}} at the Equator, to {{math|''π''}} at the South Pole, and the [[longitude]] {{mvar|φ}}, or [[azimuth]], may assume all values with {{math|0 ≤ ''φ'' < 2''π''}}.  For a fixed integer {{mvar|ℓ}}, every solution {{math|''Y''(''θ'', ''φ'')}} of the eigenvalue problem
<math display="block"> r^2\nabla^2 Y = -\ell (\ell + 1 ) Y</math>
is a [[linear combination]] of {{math|''Y<sub>ℓ</sub><sup>m</sup>''}}.  In fact, for any such solution, {{math|''r<sup>ℓ</sup> Y''(''θ'', ''φ'')}} is the expression in spherical coordinates of a [[homogeneous polynomial]] that is harmonic (see [[Spherical harmonics#Higher dimensions|below]]), and so counting dimensions shows that there are {{math|2''ℓ'' + 1}} linearly independent such polynomials.

The general solution to Laplace's equation in a ball centered at the origin is a [[linear combination]] of the spherical harmonic functions multiplied by the appropriate scale factor {{math|''r<sup>ℓ</sup>''}},
<math display="block"> f(r, \theta, \varphi) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m r^\ell Y_\ell^m (\theta, \varphi ), </math>
where the {{math|''f<sub>ℓ</sub><sup>m</sup>''}} are constants and the factors {{math|''r<sup>ℓ</sup> Y<sub>ℓ</sub><sup>m</sup>''}} are known as [[solid harmonics]]. Such an expansion is valid in the [[Ball (mathematics)|ball]]
<math display="block"> r < R = \frac{1}{\limsup_{\ell\to\infty} |f_\ell^m|^{{1}/{\ell}}}.</math>

For <math> r > R</math>, the solid harmonics with negative powers of <math>r</math> are chosen instead. In that case, one needs to expand the solution of known regions in [[Laurent series]] (about <math>r=\infty</math>), instead of [[Taylor series]] (about <math>r = 0</math>), to match the terms and find <math>f^m_\ell</math>.