# 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 Yl is the circular harmonic. It is expressed as

$Y_{l}(\theta )=P_{l}(cos\theta )$ where Pl 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·Ylm(θ,φ)
then the representative surface looks like a "battered" sphere;
Ylm is equal to 0 along circles (the representative surface intersects the ρ = ρ0 sphere at these circles). Ylm is alternatively positive and negative between two circles.