# 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)

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:

$l\,$ Cartesian plot of $Y_l^0\left( \theta \right)$ Polar plot of $r_0+r_1Y_l^0\left( \theta \right)$ Polar plot of $|Y_l^0\left( \theta \right)|^2$
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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.
the Y32 with four sections