Spherical harmonic

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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)[edit]

A function is decomposed as

where Yl is the circular harmonic. It is expressed as

where Pl is the Legendre polynomial

The circular harmonics are represented in three ways:

Cartesian plot of Polar plot of Polar plot of
1 Legendre(1,0) Cos(th).svg Circular Harmonic Y(1,0) Polar Diff.svg Circular Harmonic Y(1,0)^2 Polar.svg
2 Legendre(2,0) Cos(th).svg Circular Harmonic Y(2,0) Polar Diff.svg Circular Harmonic Y(2,0)^2 Polar.svg
3 Legendre(3,0) Cos(th).svg Circular Harmonic Y(3,0) Polar Diff.svg Circular Harmonic Y(3,0)^2 Polar.svg
4 Legendre(4,0) Cos(th).svg

In space[edit]

  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.
the Y32 with four sections