# 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 *Y*_{l} is the circular harmonic. It is expressed as

where *P*_{l} 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 | |||

2 | |||

3 | |||

4 |

## In space

[edit]m=0 | m=1 | m=2 | m=3 | m=4 | |
---|---|---|---|---|---|

l=0 | |||||

l=1 | |||||

l=2 | |||||

l=3 | |||||

l=4 |