Legendre polynomials

Associated Legendre polynomials are the colatitudinal part of the spherical harmonics which are common to all separations of Laplace's equation in spherical polar coordinates. The radial part of the solution varies from one potential to another, but the harmonics are always the same and are a consequence of spherical symmetry. Associated polynomials have to be used when the solutions have an azimuthal component $e^{2m\pi i}$, for which reason the term dependent on m appears in the differential equation:

 

$$\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right) -\frac{m^2}{\sin^2\theta}\Theta +\ell(\ell+1)\Theta$$ = 0. (325)

  

Figure 17: A spherical harmonic embedded in a contour map.

Figure 17 shows a visual representation of the spherical harmonic with four azimuthal nodes and two colatitudinal nodes. Each spherical harmonic subdivides the surface of a sphere into squares analogous to the action of $\sin(x)\sin(y)$ in the plane, except that they have to fit onto the surface of a sphere and respect the averaging principle which makes functions harmonic. Quite general functions defined over the sphere can be constructed as linear combinations, end even infinite series, of spherical harmonics.

Generally speaking, the solutions of equation (325) are only polynomials for integer values of $\ell$ and m, and even then it may be only one of the two linearly independent solutions which is a polynomial, but the full implications of this situation will only become fully apparent when the spectral density is discussed. In any event, the coefficients in the differential equation become singular at the poles, where the colatitude is either 0 or $\pi$, so especial attention must be given to the solutions at these points.

Even though the associated Legendre functions satisfy a real differential equation, the possibility esists of treating either $\ell$, or m, or both, as complex variables, with respect to which continuation into the complex plane is possible.