Spherical Harmonic Closure Relations

Author

Eric W. Weisstein
December 27, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/SphericalHarmonicClosureRelations.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SphericalHarmonicClosureRelations.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Sums

(∑_(m=-l)^l)((Abs[(Subscript[Y, l]^m)(θ,ϕ)])^2)==((2 l+1)/(4 π))

TimeConstrained[Sum[((Abs[SphericalHarmonicY[l,m,θ,ϕ]])^2),{m,-l,l}],300]//Timing

Subtract@@@Table[
With[{l=RandomInteger[{1,10}],θ=RandomReal[{0,2π}],ϕ=RandomReal[{0,π}]},
{Sum[((Abs[SphericalHarmonicY[l,m,θ,ϕ]])^2),{m,-l,l}],(2l+1)/(4π)}],{100}]//Chop//Union

(∑_(m=-l)^l)(Subscript[Y, l]^m)(θ,ϕ) (Conjugate[(Subscript[Y, l]^m)(θ,ϕ)])==((2 l+1)/(4 π))

TimeConstrained[Sum[SphericalHarmonicY[l,m,θ,ϕ]Conjugate[SphericalHarmonicY[l,m,θ,ϕ]],{m,-l,l}],300]//Timing

Subtract@@@Table[
With[{l=RandomInteger[{1,10}],θ=RandomReal[{0,2π}],ϕ=RandomReal[{0,π}]},
{Sum[SphericalHarmonicY[l,m,θ,ϕ]Conjugate[SphericalHarmonicY[l,m,θ,ϕ]],{m,-l,l}],(2l+1)/(4π)}],{100}]//Chop//Union

(∑_(l=0)^∞)(∑_(m=-l)^l)(Subscript[Y, l]^m)(θ1,ϕ1) (Conjugate[(Subscript[Y, l]^m)(θ2,ϕ2)])==(DiracDeltaSeq[ϕ1-ϕ2]) (DiscreteDeltaSeq[cos(θ1)-cos(θ2)])

Sum[SphericalHarmonicY[l,m,θ1,ϕ1]Conjugate[SphericalHarmonicY[l,m,θ2,ϕ2]],{l,0,∞},{m,-l,l}]

Table[
With[{θ1=RandomReal[{0,2π}],ϕ1=RandomReal[{0,π}],θ2=RandomReal[{0,2π}],ϕ2=RandomReal[{0,π}]},
Sum[SphericalHarmonicY[l,m,θ1,ϕ1]Conjugate[SphericalHarmonicY[l,m,θ2,ϕ2]],{l,0,500},{m,-l,l}]],{1}]//Chop//Union