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:
- in cartesian coordinates:
- in polar coordinates:
- in polar coordinates:
Cartesian plot of | Polar plot of | Polar plot of | |
---|---|---|---|
1 | ![]() |
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2 | ![]() |
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3 | ![]() |
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4 | ![]() |
In space
[edit]m=0 | m=1 | m=2 | m=3 | m=4 | |
---|---|---|---|---|---|
l=0 | |||||
l=1 | |||||
l=2 | |||||
l=3 | |||||
l=4 |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Harmoniques_spheriques_positif_negatif.png/300px-Harmoniques_spheriques_positif_negatif.png)
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.
![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/08/Traces_harmonique_spherique.png/300px-Traces_harmonique_spherique.png)