Table 1.

Explicit expressions of the spherical harmonics up to degree 3, in both polar and Cartesian form due to Brechbühler. The last part of the table gives the common normalizing constants, e.g. $Y_{0}^{1} = \sqrt{3 ∕ 4 π u_{2}}$

l	m = 0	m = 1	m = 2	m = 3
0	1
1	cosθ	e^iφ sinθ
2	-1+3cos²θ	e^iφ cosθ sinθ	e^2iφ sin²θ
3	-3cosθ+5cos³θ	e^iφ (1-5cos²θ) sinθ	e^2iφ cosθsin²θ	e^3iφsin³θ

0	1
1	u ₂	u₀+iu₁
2	-1+3u₂²	(u₀+iu₁)u₂	(u₀+iu₁)²
3	-3u₂+5u₂³	(u₀+iu₁)(1-5u₂²)	(u₀+iu₁)²u₂	(u₀+iu₁)³

0	$1 ∕ \sqrt{4 π}$
1	$\sqrt{3 ∕ 4 π}$	$\sqrt{3 ∕ 8 π}$
2	$\sqrt{5 ∕ 16 π}$	$\sqrt{15 ∕ 8 π}$	$\sqrt{15 ∕ 32 π}$
3	$\sqrt{7 ∕ 16 π}$	$\sqrt{21 ∕ 64 π}$	$\sqrt{105 ∕ 32 π}$	$\sqrt{35 ∕ 64 π}$