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

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

where Yl is the circular harmonic. It is expressed as

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

where Pl is the Legendre polynomial

The circular harmonics are represented in three ways:

• in cartesian coordinates: ${\displaystyle y=Y_{l}(\theta )}$
• in polar coordinates: ${\displaystyle r=r_{0}+r_{1}\cdot Y_{l}(\theta )}$
• in polar coordinates: ${\displaystyle r=|Y_{l}(\theta )|^{2}}$
${\displaystyle l\,}$ Cartesian plot of ${\displaystyle Y_{l}^{0}\left(\theta \right)}$ Polar plot of ${\displaystyle r_{0}+r_{1}Y_{l}^{0}\left(\theta \right)}$ Polar plot of ${\displaystyle |Y_{l}^{0}\left(\theta \right)|^{2}}$
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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