Spherical Harmonics

The spherical harmonics are special functions on the 2-sphere $\mathbb S^2$. In terms of polar coordinates ${\bf \xi} = (\sin \theta \cos \rho, \sin \theta \sin \rho, \cos \theta)$ the spherical harmonic of degree $m$ and order $l$ is defined by

$$Y_m^l({\bf \xi}) = \sqrt{\frac{2m+1}{4\pi}} \, P_m^{|l|}(\cos\theta) \, \mathrm e^{\mathrm i l\rho}$$

where $P_m^{|l|}\colon[-1,1]\to\mathbb R$, $m \in {\bf N_0}$, and $l = -m, \ldots m$ denote the associated Legendre-Polynomials defined by

$$P_m^l(x) = \sqrt{(m-l)!}{(m+l)!} \, (1-x^2)^{l/2} \frac{d^l}{dx^l} P_m(x)$$

and $P_m\colon[-1,1]\to\mathbb R$ denotes the Legendre polynomials given by their corresponding Rodrigues formula

$$P_m(x) = \frac{1}{2^m\,m!} \, \frac{d^m}{dx^m}(x^2-1)^m.$$

Hence in MTEX the spherical harmonics are normalized with respect to the $L^2(\mathbb S^2)$ norm.

We get the function values of the spherical harmonics of degree 1 in a point v by the command sphericalY, i.e.

v = vector3d.X
sphericalY(1,v)

v = vector3d (y↑→x)
  x y z
  1 0 0
ans =
    0.3455    0.0000    0.3455

The spherical harmonics form an orthonormal basis in $L_2(\mathbb S^2)$. Hence we describe functions on the 2-sphere by there harmonic representation using the class S2FunHarmonic.

With that we define the spherical harmonic $Y_1^1$ by

Y = S2FunHarmonic([0;0;0;1])
Y.eval(v)

Y = S2FunHarmonic (y↑→x)
  bandwidth: 1
 
ans =
    0.3455

Various normalizations for the sperical harmonics are common in the literature.

Here we define the $L_2$-norm by

$$\| f \|_2 = \left(\int_{\mathrm{sphere}} \lvert f(\xi)\rvert^2 \,\mathrm d\xi\right)^{1/2}$$

such that $\| 1 \|_2^2 = 1$. Take a look on the section Integration of S2Fun's.

Using that definition the spherical harmonics in MTEX fulfill

$$\| Y_m^l \|_2 = 1$$ for all $m,l$.

norm(Y)

ans =
     1

To concluse this section we plot the first ten spherical harmonics

surf(S2FunHarmonic(eye(10)))