Editing Associated Legendre polynomials (section)

==Applications in physics: spherical harmonics==
{{main|Spherical harmonics}}
In many occasions in [[physics]], associated Legendre polynomials in terms of angles occur where [[spherical]] [[symmetry]] is involved.  The colatitude angle in [[spherical coordinates]] is
the angle <math>\theta</math> used above.  The longitude angle, <math>\phi</math>, appears in a multiplying factor.  Together, they make a set of functions called [[spherical harmonic]]s. These functions express the symmetry of the [[Riemann sphere|two-sphere]] under the action of the [[Lie group]] SO(3).{{cn|date=July 2022}}

What makes these functions useful is that they are central to the solution of the equation
<math>\nabla^2\psi + \lambda\psi = 0</math> on the surface of a sphere.  In spherical coordinates θ (colatitude) and φ (longitude), the [[Laplacian]] is

<math display="block">\nabla^2\psi = \frac{\partial^2\psi}{\partial\theta^2} + \cot \theta \frac{\partial \psi}{\partial \theta} + \csc^2 \theta\frac{\partial^2\psi}{\partial\phi^2}.</math>

When the [[partial differential equation]]

<math display="block">\frac{\partial^2\psi}{\partial\theta^2} + \cot \theta \frac{\partial \psi}{\partial \theta} + \csc^2 \theta\frac{\partial^2\psi}{\partial\phi^2} + \lambda \psi = 0</math>

is solved by the method of [[separation of variables#pde|separation of variables]], one gets a φ-dependent part <math>\sin(m\phi)</math> or <math>\cos(m\phi)</math> for integer m≥0, and an equation for the θ-dependent part

<math display="block">\frac{d^{2}y}{d\theta^2} + \cot \theta \frac{dy}{d\theta} + \left[\lambda - \frac{m^2}{\sin^2\theta}\right]\,y = 0\,</math>

for which the solutions are <math>P_\ell^{m}(\cos \theta)</math> with <math>\ell{\ge}m</math>
and <math>\lambda = \ell(\ell+1)</math>.

Therefore, the equation

<math display="block">\nabla^2\psi + \lambda\psi = 0</math>

has nonsingular separated solutions only when <math>\lambda = \ell(\ell+1)</math>,
and those solutions are proportional to

<math display="block">P_\ell^{m}(\cos \theta)\ \cos (m\phi)\ \ \ \ 0 \le m \le \ell</math>

and

<math display="block">P_\ell^{m}(\cos \theta)\ \sin (m\phi)\ \ \ \ 0 < m \le \ell.</math>

For each choice of ''ℓ'', there are {{nowrap|2ℓ + 1}} functions
for the various values of ''m'' and choices of sine and cosine.
They are all orthogonal in both ''ℓ'' and ''m'' when integrated over the
surface of the sphere.

The solutions are usually written in terms of [[complex exponential]]s:

<math display="block">Y_{\ell, m}(\theta, \phi) =  \sqrt{\frac{(2\ell+1)(\ell-m)!}{4\pi(\ell+m)!}}\ P_\ell^{m}(\cos \theta)\ e^{im\phi}\qquad -\ell \le m \le \ell.
</math>
The functions <math>Y_{\ell, m}(\theta, \phi)</math> are the [[spherical harmonics]], and the quantity in the square root is a normalizing factor.
Recalling the relation between the associated Legendre functions of positive and negative ''m'', it is easily shown that the spherical harmonics satisfy the identity<ref>This identity can also be shown by relating the spherical harmonics to [[Wigner D-matrix|Wigner D-matrices]] and use of the time-reversal property of the latter. The relation between associated Legendre functions of ±''m'' can then be proved from the complex conjugation identity of the spherical harmonics.</ref>

<math display="block">Y_{\ell, m}^*(\theta, \phi) = (-1)^m Y_{\ell, -m}(\theta, \phi).</math>

The spherical harmonic functions form a complete orthonormal set of functions in the sense of [[Fourier series]]. Workers in the fields of geodesy, geomagnetism and spectral analysis use a different phase and normalization factor than given here (see [[spherical harmonics]]).

When a 3-dimensional spherically symmetric partial differential equation is solved by the method of separation of variables in spherical coordinates, the part that remains after removal of the radial part is typically
of the form

<math display="block">\nabla^2\psi(\theta, \phi) + \lambda\psi(\theta, \phi) = 0,</math>

and hence the solutions are spherical harmonics.