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Description

Legendre Polynomial of cos(θ) with l=3, m=0

Date

24 May 2007

Source

Own work

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Author

Inductiveload

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Category:Self-published work#Legendre(3,0)%20Cos(th).svg Category:PD-self#Legendre(3,0)%20Cos(th).svg

Mathematical Function Plot
Description	Legendre Polynomial of cos(θ) with l=3, m=0
Equation	$Y_{3}^{0}\left(\theta \right)={\frac {5}{2}}\cos ^{3}\left(\theta \right)-{\frac {3}{2}}\cos \left(\theta \right)$
Co-ordinate System	Cartesian
X Range	0 .. 2π
Y Range	-1 .. 1

Points of Interest in this Range
Maxima	$\left(0,1\right)\,,$ $\left(2\pi ,1\right)\,,$ $\cos \left(\theta \right)=-{\frac {1}{\sqrt {5}}}\,$
Minima	$\left(\pi ,-1\right)\,,$ $\cos \left(\theta \right)={\frac {1}{\sqrt {5}}}\,$
Roots	$\theta ={\frac {\pi }{2}},{\frac {3\pi }{2}},$ $\cos \left(\theta \right)=\pm {\sqrt {\frac {3}{5}}}\,$

Category:Spherical harmonics

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File:Legendre(3,0) Cos(th).svg