# The Other Spherical Harmonics

Everybody who’s anybody knows about the Legendre Polynomials $P_l$ and their associated functions $P_l^m$. It is also well-known that the associated polynomials as a function of the polar angle, $P_l^m(\cos(\theta))$, when multiplied by a complex exponential in the azimuthal angle, $e^{i\frac{m \phi}{2 \pi}}$, produce particularly important type of function known as a “Spherical Harmonic”, $Y_l^m(\theta,\phi) = P_l^m(\cos(\theta)) e^{i\frac{m \phi}{2 \pi}}.$

But what most people don’t know, is that the Legendre Polynomials actually have an evil misunderstood twin: The Legendre Q Functions

## Legendre Equation

The Legendre equation pops up in a number of different places in Physics, most notably in the contexts of electrostatic potentials and wavefunctions of the hydrogen atom. In both of those cases, the generalized legendre equation arises when using Separation of Variables in spherical coordinates to solve 2nd order PDEs – the Laplace equation in electrostatics, and the Schrödinger equation in quantum mechanics (Fun fact: In the absence of a potential function V, the Schrödinger equation reduces to a diffusion equation with an imaginary coefficient, which yields wave-like solutions with a non-trivial dispersion relation). The generalized Legendre equation is:

In cases where we can set m = 0 (As it does in separation of variables problems with azimuthal symmetry), this reduces to the Legendre equation (not generalized):

If we assume polynomial solutions – that is, of the form $P_l(x) = \sum_{j=0}^{\infty} a_j x^j$ – Then one finds a set of solutions to the Legendre equation known as the Legendre Polynomials, which are given by Rodrigues’ Formula:

(Another fun fact: Using the gamma function and fractional derivatives, Rodrigues’ formula can be extended to define $P_l$ for all real values of l, not just the integers!)

In much the same way as harmonic sinusoids in a Fourier series, The Legendre polynomials form a complete, orthogonal basis of functions in the range [-1,1]. Mathematica makes it easy to look at these functions and get some intuition about their behavior. Below (will be) two interactive plots. One which allows you to look at different polynomials $y(x) = P_l(x)$ in cartesian coordinates, and one which allows you to plot the corresponding angular functions $r(\theta) = P_l(\cos(\theta))$ in polar coordinates:

I’m running into a really weird rendering issue currently, so until I can get this to embed properly, a link to the cloud upload will have to suffice: Legendre Polynomials Interactive