Spherical harmonic

Let us consider continuous functions that only depend on the orientation in space (θ,φ). The spherical harmonics are a basis of such functions.

The decomposition in spherical harmonics is used to represent these functions ; it is similar to the Fourier transform for periodic functions.

In the plane (circular harmonics)

A function is decomposed as

$f(\theta )=\sum _{l=0}^{\infty }C_{l}\cdot Y_{l}(\theta )$

where Y_l is the circular harmonic. It is expressed as

$Y_{l}(\theta )=P_{l}(cos\theta )$

where P_l is the Legendre polynomial

The circular harmonics are represented in three ways:

• in cartesian coordinates: $y=Y_{l}(\theta )$
• in polar coordinates: $r=r_{0}+r_{1}\cdot Y_{l}(\theta )$
• in polar coordinates: $r=|Y_{l}(\theta )|^{2}$

$l\,$	Cartesian plot of $Y_{l}^{0}\left(\theta \right)$	Polar plot of $r_{0}+r_{1}Y_{l}^{0}\left(\theta \right)$	Polar plot of $|Y_{l}^{0}\left(\theta \right)|^{2}$
1
2
3
4

In space

	m=0	m=1	m=2	m=3	m=4
l=0
l=1
l=2
l=3
l=4

Representation as ρ = ρ₀ + ρ₁·Y_l^m(θ,φ)
then the representative surface looks like a "battered" sphere;
Y_l^m is equal to 0 along circles (the representative surface intersects the ρ = ρ₀ sphere at these circles). Y_l^m is alternatively positive and negative between two circles.

the Y₃² with four sections